Electronic Communications in Probability Articles (Project Euclid)
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The latest articles from Electronic Communications in Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Tue, 14 Feb 2017 04:00 ESTTue, 14 Feb 2017 04:00 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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The Intrinsic geometry of some random manifolds
http://projecteuclid.org/euclid.ecp/1483585770
<strong>Sunder Ram Krishnan</strong>, <strong>Jonathan E. Taylor</strong>, <strong>Robert J. Adler</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.
</p>projecteuclid.org/euclid.ecp/1483585770_20170214040037Tue, 14 Feb 2017 04:00 ESTFinitely dependent cycle coloringhttps://projecteuclid.org/euclid.ecp/1536977437<strong>Alexander E. Holroyd</strong>, <strong>Tom Hutchcroft</strong>, <strong>Avi Levy</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We construct stationary finitely dependent colorings of the cycle which are analogous to the colorings of the integers recently constructed by Holroyd and Liggett. These colorings can be described by a simple necklace insertion procedure, and also in terms of an Eden growth model on a tree. Using these descriptions we obtain simpler and more direct proofs of the characterizations of the 1- and 2-color marginals.
</p>projecteuclid.org/euclid.ecp/1536977437_20181029220647Mon, 29 Oct 2018 22:06 EDTOn the maximum of the discretely sampled fractional Brownian motion with small Hurst parameterhttps://projecteuclid.org/euclid.ecp/1537257726<strong>Konstantin Borovkov</strong>, <strong>Mikhail Zhitlukhin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
We show that the distribution of the maximum of the fractional Brownian motion $B^H$ with Hurst parameter $H\to 0$ over an $n$-point set $\tau \subset [0,1]$ can be approximated by the normal law with mean $\sqrt{\ln n} $ and variance $1/2$ provided that $n\to \infty $ slowly enough and the points in $\tau $ are not too close to each other.
</p>projecteuclid.org/euclid.ecp/1537257726_20181029220647Mon, 29 Oct 2018 22:06 EDTOccupation time of Lévy processes with jumps rational Laplace transformshttps://projecteuclid.org/euclid.ecp/1539137386<strong>Ait-Aoudia Djilali</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
We are interested in occupation times of Lévy processes with jumps rational Laplace transforms. The corresponding boundary value problems via the Feynman-Kac representation are solved to obtain an explicit formula for the joint distribution of the occupation time and the terminal value of the Lévy processes with jumps rational Laplace transforms.
</p>projecteuclid.org/euclid.ecp/1539137386_20181029220647Mon, 29 Oct 2018 22:06 EDTOn the maximum of conditioned random walks and tightness for pinning modelshttps://projecteuclid.org/euclid.ecp/1539309733<strong>Francesco Caravenna</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
We consider real random walks with finite variance. We prove an optimal integrability result for the diffusively rescaled maximum, when the walk or its bridge is conditioned to stay positive, or to avoid zero. As an application, we prove tightness under diffusive rescaling for general pinning and wetting models based on random walks.
</p>projecteuclid.org/euclid.ecp/1539309733_20181029220647Mon, 29 Oct 2018 22:06 EDTEigenvectors of non normal random matriceshttps://projecteuclid.org/euclid.ecp/1539309734<strong>Florent Benaych-Georges</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e} ^{-n\operatorname{Tr} V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors $\mathbf{v} ,\mathbf{v} '$ associated with distinct eigenvalues $\lambda ,\lambda '$ that are the closest to specified points $z,z'$ in the complex plane, the rescaled inner product \[ \sqrt{n} (\lambda '-\lambda )\langle \mathbf{v} ,\mathbf{v} '\rangle \] is uniformly sub-Gaussian, and give a more precise statement in the case of the Ginibre ensemble.
</p>projecteuclid.org/euclid.ecp/1539309734_20181029220647Mon, 29 Oct 2018 22:06 EDTUniform Hausdorff dimension result for the inverse images of stable Lévy processeshttps://projecteuclid.org/euclid.ecp/1539914641<strong>Renming Song</strong>, <strong>Yimin Xiao</strong>, <strong>Xiaochuan Yang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $\alpha $-stable Lévy processes with $1< \alpha \le 2$. This extends a theorem of Kaufman [11] for Brownian motion. Our method is different from that of [11] and depends on covering principles for Markov processes.
</p>projecteuclid.org/euclid.ecp/1539914641_20181029220647Mon, 29 Oct 2018 22:06 EDTConcentration inequalities for polynomials of contracting Ising modelshttps://projecteuclid.org/euclid.ecp/1539914642<strong>Reza Gheissari</strong>, <strong>Eyal Lubetzky</strong>, <strong>Yuval Peres</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the concentration of a degree-$d$ polynomial of the $N$ spins of a general Ising model, in the regime where single-site Glauber dynamics is contracting. For $d=1$, Gaussian concentration was shown by Marton (1996) and Samson (2000) as a special case of concentration for convex Lipschitz functions, and extended to a variety of related settings by e.g., Chazottes et al. (2007) and Kontorovich and Ramanan (2008). For $d=2$, exponential concentration was shown by Marton (2003) on lattices. We treat a general fixed degree $d$ with $O(1)$ coefficients, and show that the polynomial has variance $O(N^d)$ and, after rescaling it by $N^{-d/2}$, its tail probabilities decay as $\exp (- c\,r^{2/d})$ for deviations of $r \geq C \log N$.
</p>projecteuclid.org/euclid.ecp/1539914642_20181029220647Mon, 29 Oct 2018 22:06 EDTAn improved upper bound for the critical value of the contact process on $\mathbb{Z} ^d$ with $d\geq 3$https://projecteuclid.org/euclid.ecp/1539914643<strong>Xiaofeng Xue</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
By coupling the basic contact process with a linear system, we give an improved upper bound for the critical value $\lambda _c$ of the basic contact process on the lattice $\mathbb{Z} ^d$ with $d\geq 3$. As a direct corollary of our result, the critical value of the three-dimensional contact process is shown to be at most $0.34$.
</p>projecteuclid.org/euclid.ecp/1539914643_20181029220647Mon, 29 Oct 2018 22:06 EDTMean-field limit of a particle approximation of the one-dimensional parabolic-parabolic Keller-Segel model without smoothinghttps://projecteuclid.org/euclid.ecp/1540865165<strong>Jean-François Jabir</strong>, <strong>Denis Talay</strong>, <strong>Milica Tomašević</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
In this work, we prove the well–posedness of a singularly interacting stochastic particle system and we establish propagation of chaos result towards the one-dimensional parabolic-parabolic Keller-Segel model.
</p>projecteuclid.org/euclid.ecp/1540865165_20181029220647Mon, 29 Oct 2018 22:06 EDTNew characterizations of the $S$ topology on the Skorokhod spacehttps://projecteuclid.org/euclid.ecp/1515467250<strong>Adam Jakubowski</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 16 pp..</p><p><strong>Abstract:</strong><br/>
The $S$ topology on the Skorokhod space was introduced by the author in 1997 and since then it has proved to be a useful tool in several areas of the theory of stochastic processes. The paper brings complementary information on the $S$ topology. It is shown that the convergence of sequences in the $S$ topology admits a closed form description, exhibiting the locally convex character of the $S$ topology. Morover, it is proved that the $S$ topology is, up to some technicalities, finer than any linear topology which is coarser than Skorokhod’s $J_1$ topology. The paper contains also definitions of extensions of the $S$ topology to the Skorokhod space of functions defined on $[0,+\infty )$ and with multidimensional values.
</p>projecteuclid.org/euclid.ecp/1515467250_20181122220333Thu, 22 Nov 2018 22:03 ESTRespondent-Driven Sampling and Sparse Graph Convergencehttps://projecteuclid.org/euclid.ecp/1517626933<strong>Siva Athreya</strong>, <strong>Adrian Röllin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider a particular respondent-driven sampling procedure governed by a graphon. Using a specific clumping procedure of the sampled vertices, we construct a sequence of sparse graphs. If the sequence of the vertex-sets is stationary, then the sequence of sparse graphs converges to the governing graphon in the cut-metric. The tools used are a concentration inequality for Markov chains and the Stein-Chen method.
</p>projecteuclid.org/euclid.ecp/1517626933_20181122220333Thu, 22 Nov 2018 22:03 ESTOn recurrence of the multidimensional Lindley processhttps://projecteuclid.org/euclid.ecp/1518426010<strong>Wojciech Cygan</strong>, <strong>Judith Kloas</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
A Lindley process arises from classical studies in queueing theory and it usually reflects waiting times of customers in single server models. In this note we study recurrence of its higher dimensional counterpart under some mild assumptions on the tail behaviour of the underlying random walk. There are several links between the Lindley process and the associated random walk and we build upon such relations. We apply a method related to discrete subordination for random walks on the integer lattice together with various facts from the theory of fluctuations of random walks.
</p>projecteuclid.org/euclid.ecp/1518426010_20181122220333Thu, 22 Nov 2018 22:03 ESTThe Hammersley-Welsh bound for self-avoiding walk revisitedhttps://projecteuclid.org/euclid.ecp/1518426011<strong>Tom Hutchcroft</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
The Hammersley-Welsh bound ( Quart. J. Math., 1962) states that the number $c_n$ of length $n$ self-avoiding walks on $\mathbb{Z} ^d$ satisfies \[ c_n \leq \exp \left [ O(n^{1/2}) \right ] \mu _c^n, \] where $\mu _c=\mu _c(d)$ is the connective constant of $\mathbb{Z} ^d$. While stronger estimates have subsequently been proven for $d\geq 3$, for $d=2$ this has remained the best rigorous, unconditional bound available. In this note, we give a new, simplified proof of this bound, which does not rely on the combinatorial analysis of unfolding. We also prove a small, non-quantitative improvement to the bound, namely \[ c_n \leq \exp \left [ o(n^{1/2})\right ] \mu _c^n. \] The improved bound is obtained as a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond ( Commun. Math. Phys., 2013). We also show that any quantitative form of that theorem would yield a corresponding quantitative improvement to the Hammersley-Welsh bound.
</p>projecteuclid.org/euclid.ecp/1518426011_20181122220333Thu, 22 Nov 2018 22:03 ESTHitting time and mixing time bounds of Stein’s factorshttps://projecteuclid.org/euclid.ecp/1518663615<strong>Michael C.H. Choi</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
For any discrete target distribution, we exploit the connection between Markov chains and Stein’s method via the generator approach and express the solution of Stein’s equation in terms of expected hitting time. This yields new upper bounds of Stein’s factors in terms of the parameters of the Markov chain, such as mixing time and the gradient of expected hitting time. We compare the performance of these bounds with those in the literature, and in particular we consider Stein’s method for discrete uniform, binomial, geometric and hypergeometric distribution. As another application, the same methodology applies to bound expected hitting time via Stein’s factors. This article highlights the interplay between Stein’s method, modern Markov chain theory and classical fluctuation theory.
</p>projecteuclid.org/euclid.ecp/1518663615_20181122220333Thu, 22 Nov 2018 22:03 ESTA matrix Bougerol identity and the Hua-Pickrell measureshttps://projecteuclid.org/euclid.ecp/1519182082<strong>Theodoros Assiotis</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
We prove a Hermitian matrix version of Bougerol’s identity. Moreover, we construct the Hua-Pickrell measures on Hermitian matrices, as stochastic integrals with respect to a drifting Hermitian Brownian motion and with an integrand involving a conjugation by an independent, matrix analogue of the exponential of a complex Brownian motion with drift.
</p>projecteuclid.org/euclid.ecp/1519182082_20181122220333Thu, 22 Nov 2018 22:03 ESTQuasi-invariance of countable products of Cauchy measures under non-unitary dilationshttps://projecteuclid.org/euclid.ecp/1519182083<strong>Han Cheng Lie</strong>, <strong>T.J. Sullivan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 6 pp..</p><p><strong>Abstract:</strong><br/>
Consider an infinite sequence $(U_n)_{n\in \mathbb{N} }$ of independent Cauchy random variables, defined by a sequence $(\delta _n)_{n\in \mathbb{N} }$ of location parameters and a sequence $(\gamma _n)_{n\in \mathbb{N} }$ of scale parameters. Let $(W_n)_{n\in \mathbb{N} }$ be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence $(\sigma _n\gamma _n)_{n\in \mathbb{N} }$ of scale parameters, with $\sigma _n\neq 0$ for all $n\in \mathbb{N} $. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of $(U_n)_{n\in \mathbb{N} }$ and $(W_n)_{n\in \mathbb{N} }$ are equivalent if and only if the sequence $(\left \vert{\sigma _n} \right \vert -1)_{n\in \mathbb{N} }$ is square-summable.
</p>projecteuclid.org/euclid.ecp/1519182083_20181122220333Thu, 22 Nov 2018 22:03 ESTCoupling of Brownian motions in Banach spaceshttps://projecteuclid.org/euclid.ecp/1519182084<strong>Elisabetta Candellero</strong>, <strong>Wilfrid S. Kendall</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
Consider a separable Banach space $\mathcal{W} $ supporting a non-trivial Gaussian measure $\mu $. The following is an immediate consequence of the theory of Gaussian measure on Banach spaces: there exist (almost surely) successful couplings of two $\mathcal{W} $-valued Brownian motions $\mathbf{B} $ and $\widetilde{\mathbf {B}} $ begun at starting points $\mathbf{B} (0)$ and $\widetilde{\mathbf {B}} (0)$ if and only if the difference $\mathbf{B} (0)-\widetilde{\mathbf {B}} (0)$ of their initial positions belongs to the Cameron-Martin space $\mathcal{H} _\mu $ of $\mathcal{W} $ corresponding to $\mu $. For more general starting points, can there be a “coupling at time $\infty $”, such that almost surely $\|{\mathbf {B}(t)-\widetilde {\mathbf {B}}(t)}\|_{\mathcal{W} } \to 0$ as $t\to \infty $? Such couplings exist if there exists a Schauder basis of $\mathcal{W} $ which is also a $\mathcal{H} _\mu $-orthonormal basis of $\mathcal{H} _\mu $. We propose (and discuss some partial answers to) the question, to what extent can one express the probabilistic Banach space property “Brownian coupling at time $\infty $ is always possible” purely in terms of Banach space geometry?
</p>projecteuclid.org/euclid.ecp/1519182084_20181122220333Thu, 22 Nov 2018 22:03 ESTMartingale approximations for random fieldshttps://projecteuclid.org/euclid.ecp/1524881136<strong>Peligrad Magda</strong>, <strong>Na Zhang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 9 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we provide necessary and sufficient conditions for the mean square approximation of a random field by an ortho-martingale. The conditions are formulated in terms of projective criteria. Applications are given to linear and nonlinear random fields with independent innovations.
</p>projecteuclid.org/euclid.ecp/1524881136_20181122220333Thu, 22 Nov 2018 22:03 ESTDiscrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equationshttps://projecteuclid.org/euclid.ecp/1524881137<strong>Yoshihito Kazashi</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
An implicit Euler–Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A discrete analogue of maximal $L^2$-regularity of the scheme and the discretised stochastic convolution is established, which has the same form as their continuous counterpart.
</p>projecteuclid.org/euclid.ecp/1524881137_20181122220333Thu, 22 Nov 2018 22:03 ESTOptimal stopping and the sufficiency of randomized threshold strategieshttps://projecteuclid.org/euclid.ecp/1525312854<strong>Vicky Henderson</strong>, <strong>David Hobson</strong>, <strong>Matthew Zeng</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
In a classical optimal stopping problem the aim is to maximize the expected value of a functional of a diffusion evaluated at a stopping time. This note considers optimal stopping problems beyond this paradigm. We study problems in which the value associated to a stopping rule depends on the law of the stopped process. If this value is quasi-convex on the space of attainable laws then it is well known that it is sufficient to restrict attention to the class of threshold strategies. However, if the objective function is not quasi-convex, this may not be the case. We show that, nonetheless, it is sufficient to restrict attention to mixtures of threshold strategies.
</p>projecteuclid.org/euclid.ecp/1525312854_20181122220333Thu, 22 Nov 2018 22:03 ESTLocal martingales in discrete timehttps://projecteuclid.org/euclid.ecp/1525312855<strong>Vilmos Prokaj</strong>, <strong>Johannes Ruf</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
For any discrete-time $\mathsf{P} $–local martingale $S$ there exists a probability measure $\mathsf{Q} \sim \mathsf{P} $ such that $S$ is a $\mathsf{Q} $–martingale. A new proof for this result is provided. The core idea relies on an appropriate modification of an argument by Chris Rogers, used to prove a version of the fundamental theorem of asset pricing in discrete time. This proof also yields that, for any $\varepsilon >0$, the measure $\mathsf{Q} $ can be chosen so that $\frac{\mathrm {d} \mathsf {Q}} {\mathrm{d} \mathsf{P} } \leq 1+\varepsilon $.
</p>projecteuclid.org/euclid.ecp/1525312855_20181122220333Thu, 22 Nov 2018 22:03 ESTCutoff for a stratified random walk on the hypercubehttps://projecteuclid.org/euclid.ecp/1527300061<strong>Anna Ben-Hamou</strong>, <strong>Yuval Peres</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
We consider the random walk on the hypercube which moves by picking an ordered pair $(i,j)$ of distinct coordinates uniformly at random and adding the bit at location $i$ to the bit at location $j$, modulo $2$. We show that this Markov chain has cutoff at time $\frac{3} {2}n\log n$ with window of size $n$, solving a question posed by Chung and Graham (1997).
</p>projecteuclid.org/euclid.ecp/1527300061_20181122220333Thu, 22 Nov 2018 22:03 ESTComparison inequalities for suprema of bounded empirical processeshttps://projecteuclid.org/euclid.ecp/1528358638<strong>Antoine Marchina</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 7 pp..</p><p><strong>Abstract:</strong><br/>
In this Note we provide comparison moment inequalities for suprema of bounded empirical processes. Our methods are only based on a decomposition in martingale and on comparison results concerning martingales proved by Bentkus and Pinelis.
</p>projecteuclid.org/euclid.ecp/1528358638_20181122220333Thu, 22 Nov 2018 22:03 ESTLarge deviations for the maximum of a branching random walkhttps://projecteuclid.org/euclid.ecp/1528358639<strong>Nina Gantert</strong>, <strong>Thomas Höfelsauer</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider real-valued branching random walks and prove a large deviation result for the position of the rightmost particle. The position of the rightmost particle is the maximum of a collection of a random number of dependent random walks. We characterise the rate function as the solution of a variational problem. We consider the same random number of independent random walks, and show that the maximum of the branching random walk is dominated by the maximum of the independent random walks. For the maximum of independent random walks, we derive a large deviation principle as well. It turns out that the rate functions for upper large deviations coincide, but in general the rate functions for lower large deviations do not.
</p>projecteuclid.org/euclid.ecp/1528358639_20181122220333Thu, 22 Nov 2018 22:03 ESTHarnack inequality and derivative formula for stochastic heat equation with fractional noisehttps://projecteuclid.org/euclid.ecp/1528358640<strong>Litan Yan</strong>, <strong>Xiuwei Yin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
In this note, we establish the Harnack inequality and derivative formula for stochastic heat equation driven by fractional noise with Hurst index $H\in (\frac 14,\frac 12)$. As an application, we introduce a strong Feller property.
</p>projecteuclid.org/euclid.ecp/1528358640_20181122220333Thu, 22 Nov 2018 22:03 ESTStable cylindrical Lévy processes and the stochastic Cauchy problemhttps://projecteuclid.org/euclid.ecp/1528358641<strong>Markus Riedle</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this work, we consider the stochastic Cauchy problem driven by the canonical $\alpha $-stable cylindrical Lévy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficient and necessary condition for the existence of the weak and mild solution of the stochastic Cauchy problem and establish the temporal irregularity of the solution.
</p>projecteuclid.org/euclid.ecp/1528358641_20181122220333Thu, 22 Nov 2018 22:03 ESTNonconventional random matrix productshttps://projecteuclid.org/euclid.ecp/1528509621<strong>Yuri Kifer</strong>, <strong>Sasha Sodin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
Let $\xi _1,\xi _2,...$ be independent identically distributed random variables and $F:{\mathbb R}^\ell \to SL_d({\mathbb R})$ be a Borel measurable matrix-valued function. Set $X_n=F(\xi _{q_1(n)},\xi _{q_2(n)},...,\xi _{q_\ell (n)})$ where $0\leq q_1<q_2<...<q_\ell $ are increasing functions taking on integer values on integers. We study the asymptotic behavior as $N\to \infty $ of the singular values of the random matrix product $\Pi _N=X_N\cdots X_2X_1$ and show, in particular, that (under certain conditions) $\frac 1N\log \|\Pi _N\|$ converges with probability one as $N\to \infty $. We also obtain similar results for such products when $\xi _i$ form a Markov chain. The essential difference from the usual setting appears since the sequence $(X_n,\, n\geq 1)$ is long-range dependent and nonstationary.
</p>projecteuclid.org/euclid.ecp/1528509621_20181122220333Thu, 22 Nov 2018 22:03 ESTStein’s method for nonconventional sumshttps://projecteuclid.org/euclid.ecp/1528509622<strong>Yeor Hafouta</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
We obtain almost optimal convergence rate in the central limit theorem for (appropriately normalized) “nonconventional" sums of the form $S_N=\sum _{n=1}^N (F(\xi _n,\xi _{2n},...,\xi _{\ell n})-\bar F)$. Here $\{\xi _n: n\geq 0\}$ is a sufficiently fast mixing vector process with some stationarity conditions, $F$ is bounded Hölder continuous function and $\bar F$ is a certain centralizing constant. Extensions to more general functions $F$ will be discusses, as well. Our approach here is based on the so called Stein’s method, and the rates obtained in this paper significantly improve the rates in [7]. Our results hold true, for instance, when $\xi _n=(T^nf_i)_{i=1}^\wp $ where $T$ is a topologically mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when $\{\xi _n: n\geq 0\}$ forms a stationary and exponentially fast $\phi $-mixing sequence, which, for instance, holds true when $\xi _n=(f_i(\Upsilon _n))_{i=1}^\wp $ where $\Upsilon _n$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.
</p>projecteuclid.org/euclid.ecp/1528509622_20181122220333Thu, 22 Nov 2018 22:03 ESTFurther studies on square-root boundaries for Bessel processeshttps://projecteuclid.org/euclid.ecp/1529460064<strong>Larbi Alili</strong>, <strong>Hiroyuki Matsumoto</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 9 pp..</p><p><strong>Abstract:</strong><br/>
We look at decompositions of perpetuities and apply them to the study of the distributions of hitting times of Bessel processes of two types of square root boundaries. These distributions are linked giving a new proof of some Mellin transforms results obtained by DeLong [6] and Yor [17]. Several random factorizations and characterizations of the studied distributions are established.
</p>projecteuclid.org/euclid.ecp/1529460064_20181122220333Thu, 22 Nov 2018 22:03 ESTApproximating diffusion reflections at elastic boundarieshttps://projecteuclid.org/euclid.ecp/1529546623<strong>Dirk Becherer</strong>, <strong>Todor Bilarev</strong>, <strong>Peter Frentrup</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We show a probabilistic functional limit result for one-dimensional diffusion processes that are reflected at an elastic boundary which is a function of the reflection local time. Such processes are constructed as limits of a sequence of diffusions which are discretely reflected by small jumps at an elastic boundary, with reflection local times being approximated by $\varepsilon $-step processes. The construction yields the Laplace transform of the inverse local time for reflection. Processes and approximations of this type play a role in finite fuel problems of singular stochastic control.
</p>projecteuclid.org/euclid.ecp/1529546623_20181122220333Thu, 22 Nov 2018 22:03 ESTMoment bounds for some fractional stochastic heat equations on the ballhttps://projecteuclid.org/euclid.ecp/1532505672<strong>Eulalia Nualart</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain upper and lower bounds for the moments of the solution to a class of fractional stochastic heat equations on the ball driven by a Gaussian noise which is white in time and has a spatial correlation in space of Riesz kernel type. We also consider the space-time white noise case on an interval.
</p>projecteuclid.org/euclid.ecp/1532505672_20181122220333Thu, 22 Nov 2018 22:03 ESTA 2-spine decomposition of the critical Galton-Watson tree and a probabilistic proof of Yaglom’s theoremhttps://projecteuclid.org/euclid.ecp/1532505673<strong>Yan-Xia Ren</strong>, <strong>Renming Song</strong>, <strong>Zhenyao Sun</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this note we propose a two-spine decomposition of the critical Galton-Watson tree and use this decomposition to give a probabilistic proof of Yaglom’s theorem.
</p>projecteuclid.org/euclid.ecp/1532505673_20181122220333Thu, 22 Nov 2018 22:03 ESTShort proofs in extrema of spectrally one sided Lévy processeshttps://projecteuclid.org/euclid.ecp/1535767266<strong>Loïc Chaumont</strong>, <strong>Jacek Małecki</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall’s identity for spectrally positive Lévy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall’s identity, Theorem 2.4 in [20] which gives an expression for the law of the supremum of spectrally positive Lévy processes. An analogous formula is obtained for the supremum of spectrally negative Lévy processes.
</p>projecteuclid.org/euclid.ecp/1535767266_20181122220333Thu, 22 Nov 2018 22:03 ESTA radial invariance principle for non-homogeneous random walkshttps://projecteuclid.org/euclid.ecp/1536718009<strong>Nicholas Georgiou</strong>, <strong>Aleksandar Mijatović</strong>, <strong>Andrew R. Wade</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
Consider non-homogeneous zero-drift random walks in $\mathbb{R} ^d$, $d \geq 2$, with the asymptotic increment covariance matrix $\sigma ^2 (\mathbf{u} )$ satisfying $\mathbf{u} ^{\top } \sigma ^2 (\mathbf{u} ) \mathbf{u} = U$ and $\operatorname{tr} \sigma ^2 (\mathbf{u} ) = V$ in all in directions $\mathbf{u} \in \mathbb{S} ^{d-1}$ for some positive constants $U<V$. In this paper we establish weak convergence of the radial component of the walk to a Bessel process with dimension $V/U$. This can be viewed as an extension of an invariance principle of Lamperti.
</p>projecteuclid.org/euclid.ecp/1536718009_20181122220333Thu, 22 Nov 2018 22:03 ESTOn covering paths with 3 dimensional random walkhttps://projecteuclid.org/euclid.ecp/1536718010<strong>Eviatar B. Procaccia</strong>, <strong>Yuan Zhang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$ in $\mathbb{Z} ^d$. For $d\ge 4$, it has been shown in [5] that such probability decays exponentially with respect to $N$. For $d=3$, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: $\forall \varepsilon >0,\exists c_\varepsilon >0,$ \[P\left ({\rm Trace}(\mathcal{P} )\subseteq{\rm Trace} \big (\{X_n\}_{n=0}^\infty \big ) \right )\le \exp \left (-c_\varepsilon N\log ^{-(1+\varepsilon )}(N)\right ).\]
</p>projecteuclid.org/euclid.ecp/1536718010_20181122220333Thu, 22 Nov 2018 22:03 ESTThe maximum deviation of the $\text{Sine} _\beta $ counting processhttps://projecteuclid.org/euclid.ecp/1536718011<strong>Diane Holcomb</strong>, <strong>Elliot Paquette</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the maximum of the $\text{Sine} _\beta $ counting process from its expectation. We show the leading order behavior is consistent with the predictions of log–correlated Gaussian fields, also consistent with work on the imaginary part of the log–characteristic polynomial of random matrices. We do this by a direct analysis of the stochastic sine equation, which gives a description of the continuum limit of the Prüfer phases of a Gaussian $\beta $–ensemble matrix.
</p>projecteuclid.org/euclid.ecp/1536718011_20181122220333Thu, 22 Nov 2018 22:03 ESTUniqueness of solution to scalar BSDEs with $L\exp{\left (\mu \sqrt {2\log {(1+L)}}\,\right )} $-integrable terminal valueshttps://projecteuclid.org/euclid.ecp/1536718012<strong>Rainer Buckdahn</strong>, <strong>Ying Hu</strong>, <strong>Shanjian Tang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
In [5], the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) if the terminal value is $L\exp \hskip -0.5pt{\left (\mu \sqrt{2\log {(1+L)}} \right )}\hskip -0.5pt$-integrable with the positive parameter $\mu $ being bigger than a critical value $\mu _0$. In this note, we give the uniqueness result for the preceding BSDE.
</p>projecteuclid.org/euclid.ecp/1536718012_20181122220333Thu, 22 Nov 2018 22:03 ESTAbsolute continuity of complex martingales and of solutions to complex smoothing equationshttps://projecteuclid.org/euclid.ecp/1536718013<strong>Ewa Damek</strong>, <strong>Sebastian Mentemeier</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
Let $X$ be a $\mathbb{C} $-valued random variable with the property that \[X \ \text{ has the same law as } \ \sum _{j\ge 1} T_j X_j\] where $X_j$ are i.i.d. copies of $X$, which are independent of the (given) $\mathbb{C} $-valued random variables $ (T_j)_{j\ge 1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins’ martingale with complex parameter.
</p>projecteuclid.org/euclid.ecp/1536718013_20181122220333Thu, 22 Nov 2018 22:03 ESTRandom walks in doubly random sceneryhttps://projecteuclid.org/euclid.ecp/1537495700<strong>Łukasz Treszczotko</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
We provide a random walk in random scenery representation of a new class of stable self-similar processes with stationary increments introduced recently by Jung, Owada and Samorodnitsky. In the functional limit theorem they provided only a single instance of this class arose as a limit. We construct a model in which a significant portion of processes in this new class is obtained as a limit.
</p>projecteuclid.org/euclid.ecp/1537495700_20181122220333Thu, 22 Nov 2018 22:03 ESTFast mixing of metropolis-hastings with unimodal targetshttps://projecteuclid.org/euclid.ecp/1539655259<strong>James Johndrow</strong>, <strong>Aaron Smith</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 9 pp..</p><p><strong>Abstract:</strong><br/>
A well-known folklore result in the MCMC community is that the Metropolis-Hastings algorithm mixes quickly for any unimodal target, as long as the tails are not too heavy. Although we’ve heard this fact stated many times in conversation, we are not aware of any quantitative statement of this result in the literature, and we are not aware of any quick derivation from well-known results. The present paper patches this small gap in the literature, providing a generic bound based on the popular “drift-and-minorization” framework of [19]. Our main contribution is to study two sublevel sets of the Lyapunov function and use path arguments in order to obtain a sharper bound than what can typically be obtained from multistep minorization arguments.
</p>projecteuclid.org/euclid.ecp/1539655259_20181122220333Thu, 22 Nov 2018 22:03 ESTAbout Doob’s inequality, entropy and Tchebichefhttps://projecteuclid.org/euclid.ecp/1540346602<strong>Emmanuel Rio</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this note we give upper bounds on the quantiles of the one-sided maximum of a nonnegative submartingale in the class $L\log L$ or the maximum of a submartingale in $L^p$. Our upper bounds involve the entropy in the case of nonnegative martingales in the class $L\log L$ and the $L^p$-norm in the case of submartingales in $L^p$. Starting from our results on entropy, we also improve the so-called bounded differences inequality. All the results are based on optimal bounds for the conditional value at risk of real-valued random variables.
</p>projecteuclid.org/euclid.ecp/1540346602_20181122220333Thu, 22 Nov 2018 22:03 ESTA large deviation principle for the Erdős–Rényi uniform random graphhttps://projecteuclid.org/euclid.ecp/1540346603<strong>Amir Dembo</strong>, <strong>Eyal Lubetzky</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
Starting with the large deviation principle (LDP) for the Erdős–Rényi binomial random graph ${\mathcal G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph ${\mathcal G}(n,m)$ (the uniform distribution over graphs with $n$ vertices and $m$ edges), at suitable $m=m_n$. Applying the latter LDP we find that tail decays for subgraph counts in ${\mathcal G}(n,m_n)$ are controlled by variational problems, which up to a constant shift, coincide with those studied by Kenyon et al. and Radin et al. in the context of constrained random graphs, e.g., the edge/triangle model.
</p>projecteuclid.org/euclid.ecp/1540346603_20181122220333Thu, 22 Nov 2018 22:03 ESTBlock size in Geometric($p$)-biased permutationshttps://projecteuclid.org/euclid.ecp/1540346604<strong>Irina Cristali</strong>, <strong>Vinit Ranjan</strong>, <strong>Jake Steinberg</strong>, <strong>Erin Beckman</strong>, <strong>Rick Durrett</strong>, <strong>Matthew Junge</strong>, <strong>James Nolen</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
Fix a probability distribution $\mathbf p = (p_1, p_2, \ldots )$ on the positive integers. The first block in a $\mathbf p$-biased permutation can be visualized in terms of raindrops that land at each positive integer $j$ with probability $p_j$. It is the first point $K$ so that all sites in $[1,K]$ are wet and all sites in $(K,\infty )$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$ we show that $p \log K$ converges in probability to an explicit constant as $p$ tends to 0. Additionally, we prove that if $\mathbf p$ has a stretch exponential distribution, then $K$ is infinite with positive probability.
</p>projecteuclid.org/euclid.ecp/1540346604_20181122220333Thu, 22 Nov 2018 22:03 ESTA sharp symmetrized form of Talagrand’s transport-entropy inequality for the Gaussian measurehttps://projecteuclid.org/euclid.ecp/1540346605<strong>Max Fathi</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 9 pp..</p><p><strong>Abstract:</strong><br/>
This note presents a sharp transport-entropy inequality that improves on Talagrand’s inequality for the Gaussian measure, arising as a dual formulation of the functional Santaló inequality. We also discuss some extensions and connections with concentration of measure.
</p>projecteuclid.org/euclid.ecp/1540346605_20181122220333Thu, 22 Nov 2018 22:03 ESTA renewal theorem and supremum of a perturbed random walkhttps://projecteuclid.org/euclid.ecp/1540346606<strong>Ewa Damek</strong>, <strong>Bartosz Kołodziejek</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
We study tails of the supremum of a perturbed random walk under regime which was not yet considered in the literature. Our approach is based on a new renewal theorem, which is of independent interest.
We obtain first and second order asymptotics of the solution to renewal equation under weak assumptions and we apply these results to obtain first and second order asymptotics of the tail of the supremum of a perturbed random walk.
</p>projecteuclid.org/euclid.ecp/1540346606_20181122220333Thu, 22 Nov 2018 22:03 ESTOn pathwise quadratic variation for càdlàg functionshttps://projecteuclid.org/euclid.ecp/1542942174<strong>Henry Chiu</strong>, <strong>Rama Cont</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We revisit Föllmer’s concept of quadratic variation of a càdlàg function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of càdlàg processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition which implies the Lebesgue decomposition of the pathwise quadratic variation as a result, rather than requiring it as an extra condition.
</p>projecteuclid.org/euclid.ecp/1542942174_20181122220333Thu, 22 Nov 2018 22:03 ESTHigh points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal processhttps://projecteuclid.org/euclid.ecp/1542942175<strong>Constantin Glenz</strong>, <strong>Nicola Kistler</strong>, <strong>Marius A. Schmidt</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
It has been proved by Bovier & Hartung [ Elect. J. Probab. 19 (2014)] that the maximum of a variable-speed branching Brownian motion (BBM) in the weak correlation regime converges to a randomly shifted Gumbel distribution. The random shift is given by the almost sure limit of McKean’s martingale, and captures the early evolution of the system. In the Bovier-Hartung extremal process, McKean’s martingale thus plays a role which parallels that of the derivative martingale in the classical BBM. In this note, we provide an alternative interpretation of McKean’s martingale in terms of a law of large numbers for high-points of BBM, i.e. particles which lie at a macroscopic distance from the edge. At such scales, ‘McKean-like martingales’ are naturally expected to arise in all models belonging to the BBM-universality class.
</p>projecteuclid.org/euclid.ecp/1542942175_20181122220333Thu, 22 Nov 2018 22:03 ESTA functional limit theorem for the profile of random recursive treeshttps://projecteuclid.org/euclid.ecp/1542942176<strong>Alexander Iksanov</strong>, <strong>Zakhar Kabluchko</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots , X_{[n^t]}(k))_{t\geq 0}$, for each $k\in \mathbb N$. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment. Let $Y_k(t)$ be the number of the $k$th generation individuals born at times $\leq t$ in this process. Then, it is shown that the appropriately centered and normalized vector-valued process $(Y_{1}(st),\ldots , Y_k(st))_{t\geq 0}$ converges weakly, as $s\to \infty $, to the same limiting Gaussian process as above.
</p>projecteuclid.org/euclid.ecp/1542942176_20181122220333Thu, 22 Nov 2018 22:03 ESTOn multiplication in $q$-Wiener chaoseshttps://projecteuclid.org/euclid.ecp/1515467249<strong>Aurélien Deya</strong>, <strong>René Schott</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 16 pp..</p><p><strong>Abstract:</strong><br/>
We pursue the investigations initiated by Donati-Martin [9] and Effros-Popa [10] regarding the multiplication issue in the chaoses generated by the $q$-Brownian motion ($q\in (-1,1)$), along two directions: $(i)$ We provide a fully-stochastic approach to the problem and thus make a clear link with the standard Brownian setting; $(ii)$ We elaborate on the situation where the kernels are given by symmetric functions, with application to the study of the $q$-Brownian martingales.
</p>projecteuclid.org/euclid.ecp/1515467249_20181123221011Fri, 23 Nov 2018 22:10 ESTStationary distributions of the Atlas modelhttps://projecteuclid.org/euclid.ecp/1519354833<strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
In this article we study the Atlas model, which consists of Brownian particles on $ \mathbb{R} $, independent except that the Atlas (i.e., lowest ranked) particle $ X_{(1)}(t) $ receives drift $ \gamma dt $, $ \gamma \in \mathbb{R} $. For any fixed shape parameter $ a>2\gamma _- $, we show that, up to a shift $ \frac{a} {2}t $, the entire particle system has an invariant distribution $ \nu _a $, written in terms an explicit Radon-Nikodym derivative with respect to the Poisson point process of density $ ae^{a\xi } d\xi $. We further show that $ \nu _a $ indeed has the product-of-exponential gap distribution $ \pi _a $ derived in [ST17]. As a simple application, we establish a bound on the fluctuation of the Atlas particle $ X_{(1)}(t) $ uniformly in $ t $, with the gaps initiated from $ \pi _a $ and $ X_{(1)}(0)=0 $.
</p>projecteuclid.org/euclid.ecp/1519354833_20181123221011Fri, 23 Nov 2018 22:10 ESTLimiting behaviour of the stationary search cost distribution driven by a generalized gamma processhttps://projecteuclid.org/euclid.ecp/1519354834<strong>Alfred Kume</strong>, <strong>Fabrizio Leisen</strong>, <strong>Antonio Lijoiï</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
Consider a list of labeled objects that are organized in a heap. At each time, object $j$ is selected with probability $p_j$ and moved to the top of the heap. This procedure defines a Markov chain on the set of permutations which is referred to in the literature as Move-to-Front rule. The present contribution focuses on the stationary search cost, namely the position of the requested item in the heap when the Markov chain is in equilibrium. We consider the scenario where the number of objects is infinite and the probabilities $p_j$’s are defined as the normalization of the increments of a subordinator. In this setting, we provide an exact formula for the moments of any order of the stationary search cost distribution. We illustrate the new findings in the case of a generalized gamma subordinator and deal with an extension to the two–parameter Poisson–Dirichlet process, also known as Pitman–Yor process.
</p>projecteuclid.org/euclid.ecp/1519354834_20181123221011Fri, 23 Nov 2018 22:10 ESTThe lower Snell envelope of smooth functions: an optional decompositionhttps://projecteuclid.org/euclid.ecp/1519722242<strong>Erick Trevino Aguilar</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we provide general conditions under which the lower Snell envelope defined with respect to the family $\mathcal{M} $ of equivalent local-martingale probability measures of a semimartingale $S$ admits a decomposition as a stochastic integral with respect to $S$ and an optional process of finite variation. On the other hand, based on properties of predictable stopping times we establish a version of the classical backwards induction algorithm in optimal stopping for the non-linear super-additive expectation associated to $\mathcal{M} $. This result is of independent interest and we show how to apply it in order to systematically construct instances of the lower Snell envelope with no optional decomposition. Such ‘counterexamples’ strengths the scope of our conditions.
</p>projecteuclid.org/euclid.ecp/1519722242_20181123221011Fri, 23 Nov 2018 22:10 ESTColumn normalization of a random measurement matrixhttps://projecteuclid.org/euclid.ecp/1519722243<strong>Shahar Mendelson</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
In this note we answer a question of G. Lecué, by showing that column normalization of a random matrix with iid entries need not lead to good sparse recovery properties, even if the generating random variable has a reasonable moment growth. Specifically, for every $2 \leq p \leq c_1\log d$ we construct a random vector $X \in \mathbb{R} ^d$ with iid, mean-zero, variance $1$ coordinates, that satisfies $\sup _{t \in S^{d-1}} \|\bigl < {X,t} \bigr >\|_{L_q} \leq c_2\sqrt{q} $ for every $2\leq q \leq p$. We show that if $m \leq c_3\sqrt{p} d^{1/p}$ and $\tilde{\Gamma } :\mathbb{R} ^d \to \mathbb{R} ^m$ is the column-normalized matrix generated by $m$ independent copies of $X$, then with probability at least $1-2\exp (-c_4m)$, $\tilde{\Gamma } $ does not satisfy the exact reconstruction property of order $2$.
</p>projecteuclid.org/euclid.ecp/1519722243_20181123221011Fri, 23 Nov 2018 22:10 ESTThe greedy walk on an inhomogeneous Poisson processhttps://projecteuclid.org/euclid.ecp/1519722244<strong>Katja Gabrysch</strong>, <strong>Erik Thörnblad</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
The greedy walk is a deterministic walk that always moves from its current position to the nearest not yet visited point. In this paper we consider the greedy walk on an inhomogeneous Poisson point process on the real line. We prove that the property of visiting all points of the point process satisfies a $0$–$1$ law and determine explicit sufficient and necessary conditions on the mean measure of the point process for this to happen. Moreover, we provide precise results on threshold functions for the property of visiting all points.
</p>projecteuclid.org/euclid.ecp/1519722244_20181123221011Fri, 23 Nov 2018 22:10 ESTParticle approximation for Lagrangian Stochastic Models with specular boundary conditionhttps://projecteuclid.org/euclid.ecp/1519722245<strong>Mireille Bossy</strong>, <strong>Jean-François Jabir</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we prove a particle approximation, in the sense of the propagation of chaos, of a Lagrangian stochastic model submitted to specular boundary condition and satisfying the mean no-permeability condition.
</p>projecteuclid.org/euclid.ecp/1519722245_20181123221011Fri, 23 Nov 2018 22:10 ESTAsymptotic results in solvable two-charge modelshttps://projecteuclid.org/euclid.ecp/1519722246<strong>Martina Dal Borgo</strong>, <strong>Emma Hovhannisyan</strong>, <strong>Alain Rouault</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
In [16], a solvable two charge ensemble of interacting charged particles on the real line in the presence of the harmonic oscillator potential is introduced. It can be seen as a special form of a grand canonical ensemble with the total charge being fixed and unit charge particles being random. Moreover, it serves as an interpolation between the Gaussian orthogonal and the Gaussian symplectic ensembles and maintains the Pfaffian structure of the eigenvalues. A similar solvable ensemble of charged particles on the unit circle was studied in [17, 10].
In this paper we explore the sharp asymptotic behavior of the number of unit charge particles on the line and on the circle, as the total charge goes to infinity. We establish extended central limit theorems, Berry-Esseen estimates and precise moderate deviations using the machinery of the mod-Gaussian convergence developed in [6, 15, 14, 4, 7]. Also a large deviation principle is derived using the Gärtner-Ellis theorem.
</p>projecteuclid.org/euclid.ecp/1519722246_20181123221011Fri, 23 Nov 2018 22:10 ESTAlmost-sure asymptotics for the number of heaps inside a random sequencehttps://projecteuclid.org/euclid.ecp/1520391724<strong>A.-L. Basdevant</strong>, <strong>A. Singh</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
We study the minimum number of heaps required to sort a random sequence using a generalization of Istrate and Bonchis’s algorithm (2015). In a previous paper, the authors proved that the expected number of heaps grows logarithmically. In this note, we improve on the previous result by establishing the almost-sure and $L^1$ convergence.
</p>projecteuclid.org/euclid.ecp/1520391724_20181123221011Fri, 23 Nov 2018 22:10 ESTHow fast planar maps get swallowed by a peeling processhttps://projecteuclid.org/euclid.ecp/1520391725<strong>Nicolas Curien</strong>, <strong>Cyril Marzouk</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
The peeling process is an algorithmic procedure that discovers a random planar map step by step. In generic cases such as the UIPT or the UIPQ, it is known [15] that any peeling process will eventually discover the whole map. In this paper we study the probability that the origin is not swallowed by the peeling process until time $n$ and show it decays at least as $n^{-2c/3}$ where \[ c \approx 0.1283123514178324542367448657387285493314266204833984375... \] is defined via an integral equation derived using the Lamperti representation of the spectrally negative $3/2$-stable Lévy process conditioned to remain positive [12] which appears as a scaling limit for the perimeter process. As an application we sharpen the upper bound of the sub-diffusivity exponent for random walk of [4].
</p>projecteuclid.org/euclid.ecp/1520391725_20181123221011Fri, 23 Nov 2018 22:10 ESTThe Vlasov-Fokker-Planck equation in non-convex landscapes: convergence to equilibriumhttps://projecteuclid.org/euclid.ecp/1521079417<strong>Manh Hong Duong</strong>, <strong>Julian Tugaut</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.
</p>projecteuclid.org/euclid.ecp/1521079417_20181123221011Fri, 23 Nov 2018 22:10 ESTThe largest root of random Kac polynomials is heavy tailedhttps://projecteuclid.org/euclid.ecp/1521079421<strong>Raphaël Butez</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 9 pp..</p><p><strong>Abstract:</strong><br/>
We prove that the largest and smallest root in modulus of random Kac polynomials have a non-universal behavior. They do not converge towards the edge of the support of the limiting distribution of the zeros. This non-universality is surprising as the large deviations principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviations principle is non-universal but the fluctuations of the largest eigenvalue are universal. We show that the modulus of the largest zero is heavy tailed, with a number of finite moments bounded from above by the behavior at the origin of the distribution of the coefficients. We also prove that the random process of the roots of modulus smaller than one converges towards a limit point process. Finally, in the case of complex Gaussian coefficients, we use the work of Peres and Virág [15] to obtain explicit formulas for the limiting objects.
</p>projecteuclid.org/euclid.ecp/1521079421_20181123221011Fri, 23 Nov 2018 22:10 ESTOrder statistics of the moduli of the eigenvalues of product random matrices from polynomial ensembleshttps://projecteuclid.org/euclid.ecp/1522375377<strong>Yanhui Wang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
Let $X_{1}, \ldots , X_{m_{N}}$ be independent random matrices of order $N$ drawn from the polynomial ensembles of derivative type. For any fixed $n$, we consider the limiting distribution of the $n$th largest modulus of the eigenvalues of $X = \prod _{k=1}^{m_{N}}X_{k}$ as $N \to \infty $ where $m_{N}/N$ converges to some constant $\tau \in [0, \infty )$. In particular, we find that the limiting distributions of spectral radii behave like that of products of independent complex Ginibre matrices.
</p>projecteuclid.org/euclid.ecp/1522375377_20181123221011Fri, 23 Nov 2018 22:10 ESTHausdorff dimension of the record set of a fractional Brownian motionhttps://projecteuclid.org/euclid.ecp/1522375381<strong>Lucas Benigni</strong>, <strong>Clément Cosco</strong>, <strong>Assaf Shapira</strong>, <strong>Kay Jörg Wiese</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
We prove that the Hausdorff dimension of the record set of a fractional Brownian motion with Hurst parameter $H$ equals $H$.
</p>projecteuclid.org/euclid.ecp/1522375381_20181123221011Fri, 23 Nov 2018 22:10 ESTOn the ladder heights of random walks attracted to stable laws of exponent 1https://projecteuclid.org/euclid.ecp/1522375382<strong>Kôhei Uchiyama</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
Let $Z$ be the first ladder height of a one dimensional random walk $S_n=X_1+\cdots + X_n$ with i.i.d. increments $X_j$ which are in the domain of attraction of a stable law of exponent $\alpha $, $0<\alpha \leq 1$. We show that $P[Z>x]$ is slowly varying at infinity if and only if $\lim _{n\to \infty } n^{-1}\sum _1^n P[S_k>0]=0$. By a known result this provides a criterion for $S_{T(R)} /R \stackrel{{\rm P}} \longrightarrow \infty $ as $R\to \infty $, where $T(R)$ is the time when $S_n$ crosses over the level $R$ for the first time. The proof mostly concerns the case $\alpha =1$.
</p>projecteuclid.org/euclid.ecp/1522375382_20181123221011Fri, 23 Nov 2018 22:10 ESTA moment-generating formula for Erdős-Rényi component sizeshttps://projecteuclid.org/euclid.ecp/1524708114<strong>Balázs Ráth</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
We derive a simple formula characterizing the distribution of the size of the connected component of a fixed vertex in the Erdős-Rényi random graph which allows us to give elementary proofs of some results of [9] and [12] about the susceptibility in the subcritical graph and the CLT [17] for the size of the giant component in the supercritical graph.
</p>projecteuclid.org/euclid.ecp/1524708114_20181123221011Fri, 23 Nov 2018 22:10 ESTWhere does a random process hit a fractal barrier?https://projecteuclid.org/euclid.ecp/1524881133<strong>Itai Benjamini</strong>, <strong>Alexander Shamov</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 5 pp..</p><p><strong>Abstract:</strong><br/>
Given a Brownian path $\beta (t)$ on $\mathbb{R} $, starting at $1$, a.s. there is a singular time set $T_{\beta }$, such that the first hitting time of $\beta $ by an independent Brownian motion, starting at $0$, is in $T_{\beta }$ with probability one. A couple of problems regarding hitting measure for random processes are presented.
</p>projecteuclid.org/euclid.ecp/1524881133_20181123221011Fri, 23 Nov 2018 22:10 ESTChaos expansion of 2D parabolic Anderson modelhttps://projecteuclid.org/euclid.ecp/1524881134<strong>Yu Gu</strong>, <strong>Jingyu Huang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
We prove a chaos expansion for the 2D parabolic Anderson Model in small time, with the expansion coefficients expressed in terms of the annealed density function of the polymer in a white noise environment.
</p>projecteuclid.org/euclid.ecp/1524881134_20181123221011Fri, 23 Nov 2018 22:10 ESTExistence of solution to scalar BSDEs with $L\exp{\left (\!\!\sqrt {{2\over \lambda }\log {(1+L)}}\,\right )} $-integrable terminal valueshttps://projecteuclid.org/euclid.ecp/1524881135<strong>Ying Hu</strong>, <strong>Shanjian Tang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we study a scalar linearly growing backward stochastic differential equation (BSDE) with an $L\exp{\left (\!\!\sqrt {{2\over \lambda }\log {(1+L)}}\,\right )} $-integrable terminal value. We prove that a BSDE admits a solution if the terminal value satisfies the preceding integrability condition with the positive parameter $\lambda $ being less than a critical value $\lambda _0$, which is weaker than the usual $L^p$ ($p>1$) integrability and stronger than $L\log L$ integrability. We show by a counterexample that the conventionally expected $L\log L$ integrability and even the preceding integrability for $\lambda >\lambda _0$ are not sufficient for the existence of solution to a BSDE with a linearly growing generator.
</p>projecteuclid.org/euclid.ecp/1524881135_20181123221011Fri, 23 Nov 2018 22:10 ESTRandom walk on the randomly-oriented Manhattan latticehttps://projecteuclid.org/euclid.ecp/1532505674<strong>Sean Ledger</strong>, <strong>Bálint Tóth</strong>, <strong>Benedek Valkó</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
In the randomly-oriented Manhattan lattice, every line in $\mathbb{Z} ^d$ is assigned a uniform random direction. We consider the directed graph whose vertex set is $\mathbb{Z} ^d$ and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the $d$ legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.
</p>projecteuclid.org/euclid.ecp/1532505674_20181123221011Fri, 23 Nov 2018 22:10 ESTA user-friendly condition for exponential ergodicity in randomly switched environmentshttps://projecteuclid.org/euclid.ecp/1532505675<strong>Michel Benaïm</strong>, <strong>Tobias Hurth</strong>, <strong>Edouard Strickler</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed in [12] that if one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium from which one can reach a point satisfying a weak Hörmander-bracket condition, then the process converges in total variation to a unique invariant probability measure. In this note, adapting the proof in [12] and using results of [5], the assumption of a G.A.S. equilibrium is weakened to the existence of an accessible point at which a barycentric combination of the vector fields vanishes. Some examples are given which demonstrate the usefulness of this condition.
</p>projecteuclid.org/euclid.ecp/1532505675_20181123221011Fri, 23 Nov 2018 22:10 ESTOn a strong form of propagation of chaos for McKean-Vlasov equationshttps://projecteuclid.org/euclid.ecp/1532657017<strong>Daniel Lacker</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
This note shows how to considerably strengthen the usual mode of convergence of an $n$-particle system to its McKean-Vlasov limit, often known as propagation of chaos, when the volatility coefficient is nondegenerate and involves no interaction term. Notably, the empirical measure converges in a much stronger topology than weak convergence, and any fixed $k$ particles converge in total variation to their limit law as $n\rightarrow \infty $. This requires minimal continuity for the drift in both the space and measure variables. The proofs are purely probabilistic and rather short, relying on Girsanov’s and Sanov’s theorems. Along the way, some modest new existence and uniqueness results for McKean-Vlasov equations are derived.
</p>projecteuclid.org/euclid.ecp/1532657017_20181123221011Fri, 23 Nov 2018 22:10 ESTPoisson-Dirichlet statistics for the extremes of a randomized Riemann zeta functionhttps://projecteuclid.org/euclid.ecp/1532657018<strong>Frédéric Ouimet</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 15 pp..</p><p><strong>Abstract:</strong><br/>
In [4], the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a consequence, we find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is expected that we can adapt the approach to prove the same result for the Riemann zeta function itself.
</p>projecteuclid.org/euclid.ecp/1532657018_20181123221011Fri, 23 Nov 2018 22:10 ESTPerfect shuffling by lazy swapshttps://projecteuclid.org/euclid.ecp/1532657019<strong>Omer Angel</strong>, <strong>Alexander E. Holroyd</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
We characterize the minimum-length sequences of independent lazy simple transpositions whose composition is a uniformly random permutation. For every reduced word of the reverse permutation there is exactly one valid way to assign probabilities to the transpositions. It is an open problem to determine the minimum length of such a sequence when the simplicity condition is dropped.
</p>projecteuclid.org/euclid.ecp/1532657019_20181123221011Fri, 23 Nov 2018 22:10 ESTTail asymptotics of maximums on trees in the critical casehttps://projecteuclid.org/euclid.ecp/1532678499<strong>Mariusz Maślanka</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
We consider the endogenous solution to the stochastic recursion \[ X\,{\buildrel \mathit {d}\over =}\,\bigvee _{i=1}^{N}{A_iX_i}\vee{B} ,\] where $N$ is a random natural number, $B$ and $\{A_i\}_{i\in{\mathbb {N}} }$ are random nonnegative numbers and $X_i$ are independent copies of $X$, independent also of $N$, $B$, $\{A_i\}_{i\in \mathbb{N} }$. The properties of solutions to this equation are governed mainly by the function $m(s)=\mathbb{E} \left [\sum _{i=1}^{N}{A_i^s}\right ]$. Recently, Jelenković and Olvera-Cravioto assuming, inter alia, $m(s)<1$ for some $s$, proved that the asymptotic behavior of the endogenous solution $R$ to the above equation is power-law, i.e. \[\mathbb{P} [R>t]\sim{Ct^{-\alpha }} \] for some $\alpha >0$ and $C>0$. In this paper we prove an analogous result when $m(s)=1$ has unique solution $\alpha >0$ and $m(s)>1$ for all $s\not =\alpha $.
</p>projecteuclid.org/euclid.ecp/1532678499_20181123221011Fri, 23 Nov 2018 22:10 ESTNon-triviality of the vacancy phase transition for the Boolean modelhttps://projecteuclid.org/euclid.ecp/1533002443<strong>Mathew D. Penrose</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there exists a strictly positive value for the intensity such that the vacant region percolates.
</p>projecteuclid.org/euclid.ecp/1533002443_20181123221011Fri, 23 Nov 2018 22:10 ESTCritical radius and supremum of random spherical harmonics (II)https://projecteuclid.org/euclid.ecp/1535767261<strong>Renjie Feng</strong>, <strong>Xingcheng Xu</strong>, <strong>Robert J. Adler</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
We continue the study, begun in [6], of the critical radius of embeddings, via deterministic spherical harmonics, of fixed dimensional spheres into higher dimensional ones, along with the associated problem of the distribution of the suprema of random spherical harmonics. Whereas [6] concentrated on spherical harmonics of a common degree, here we extend the results to mixed degrees, en passant improving on the lower bounds on critical radii that we found previously.
</p>projecteuclid.org/euclid.ecp/1535767261_20181123221011Fri, 23 Nov 2018 22:10 ESTConvergence of maximum bisection ratio of sparse random graphshttps://projecteuclid.org/euclid.ecp/1535767262<strong>Brice Huang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
We consider sequences of large sparse random graphs whose degree distribution approaches a limit with finite mean. This model includes both the random regular graphs and the Erdös-Renyi graphs of constant average degree. We prove that the maximum bisection ratio of such a graph sequence converges almost surely to a deterministic limit. We extend this result to so-called 2-spin spin glasses in the paramagnetic to ferromagnetic regime. Our work generalizes the graph interpolation method to some non-additive graph parameters.
</p>projecteuclid.org/euclid.ecp/1535767262_20181123221011Fri, 23 Nov 2018 22:10 ESTProjections of spherical Brownian motionhttps://projecteuclid.org/euclid.ecp/1535767263<strong>Aleksandar Mijatović</strong>, <strong>Veno Mramor</strong>, <strong>Gerónimo Uribe Bravo</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R} ^{n+\ell }$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R} ^{n+\ell }$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.
</p>projecteuclid.org/euclid.ecp/1535767263_20181123221011Fri, 23 Nov 2018 22:10 ESTFluctuations for block spin Ising modelshttps://projecteuclid.org/euclid.ecp/1535767264<strong>Matthias Löwe</strong>, <strong>Kristina Schubert</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We analyze the high temperature fluctuations of the magnetization of the so-called Ising block model. This model was recently introduced by Berthet et al. in [2]. We prove a Central Limit Theorems (CLT) for the magnetization in the high temperature regime. At the same time we show that this CLT breaks down at a line of critical temperatures. At this line we prove a non-standard CLT for the magnetization.
</p>projecteuclid.org/euclid.ecp/1535767264_20181123221011Fri, 23 Nov 2018 22:10 ESTNon-convergence of proportions of types in a preferential attachment graph with three co-existing typeshttps://projecteuclid.org/euclid.ecp/1535767265<strong>John Haslegrave</strong>, <strong>Jonathan Jordan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider the preferential attachment model with multiple vertex types introduced by Antunović, Mossel and Rácz. We give an example with three types, based on the game of rock-paper-scissors, where the proportions of vertices of the different types almost surely do not converge to a limit, giving a counterexample to a conjecture of Antunović, Mossel and Rácz. We also consider another family of examples where we show that the conjecture does hold.
</p>projecteuclid.org/euclid.ecp/1535767265_20181123221011Fri, 23 Nov 2018 22:10 ESTFractional Brownian motion with zero Hurst parameter: a rough volatility viewpointhttps://projecteuclid.org/euclid.ecp/1536718014<strong>Eyal Neuman</strong>, <strong>Mathieu Rosenbaum</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when $H$ goes to zero. We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.
</p>projecteuclid.org/euclid.ecp/1536718014_20181123221011Fri, 23 Nov 2018 22:10 ESTCoalescing random walk on unimodular graphshttps://projecteuclid.org/euclid.ecp/1536804170<strong>Eric Foxall</strong>, <strong>Tom Hutchcroft</strong>, <strong>Matthew Junge</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
Coalescing random walk on a unimodular random rooted graph for which the root has finite expected degree visits each site infinitely often almost surely. A corollary is that an opinion in the voter model on such graphs has infinite expected lifetime. Additionally, we deduce an adaptation of our main theorem that holds uniformly for coalescing random walk on finite random unimodular graphs with degree distribution stochastically dominated by a probability measure with finite mean.
</p>projecteuclid.org/euclid.ecp/1536804170_20181123221011Fri, 23 Nov 2018 22:10 ESTExistence of an unbounded vacant set for subcritical continuum percolationhttps://projecteuclid.org/euclid.ecp/1536977436<strong>Daniel Ahlberg</strong>, <strong>Vincent Tassion</strong>, <strong>Augusto Teixeira</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
We consider the Poisson Boolean percolation model in $\mathbb{R} ^2$, where the radius of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in $\mathbb{R} ^d$, for any $d\ge 2$, finite moment of order $d$ is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.
</p>projecteuclid.org/euclid.ecp/1536977436_20181123221011Fri, 23 Nov 2018 22:10 ESTA Brownian optimal switching problem under incomplete informationhttps://projecteuclid.org/euclid.ecp/1537840941<strong>Marcus Olofsson</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we study an incomplete information optimal switching problem in which the manager only has access to noisy observations of the underlying Brownian motion $\{W_t\}_{t \geq 0}$. The manager can, at a fixed cost, switch between having the production facility open or closed and must find the optimal management strategy using only the noisy observations. Using the theory of linear stochastic filtering, we reduce the incomplete information problem to a full information problem, show that the value function is non-decreasing with the amount of information available, and that the value function of the incomplete information problem converges to the value function of the corresponding full information problem as the noise in the observed process tends to $0$. Our approach is deterministic and relies on the PDE-representation of the value function.
</p>projecteuclid.org/euclid.ecp/1537840941_20181123221011Fri, 23 Nov 2018 22:10 ESTA note on tail triviality for determinantal point processeshttps://projecteuclid.org/euclid.ecp/1539763345<strong>Russell Lyons</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 3 pp..</p><p><strong>Abstract:</strong><br/>
We give a very short proof that determinantal point processes have a trivial tail $\sigma $-field. This conjecture of the author has been proved by Osada and Osada as well as by Bufetov, Qiu, and Shamov. The former set of authors relied on the earlier result of the present author that the conjecture held in the discrete case, as does the present short proof.
</p>projecteuclid.org/euclid.ecp/1539763345_20181123221011Fri, 23 Nov 2018 22:10 ESTApproximation of a generalized continuous-state branching process with interactionhttps://projecteuclid.org/euclid.ecp/1539763346<strong>Ibrahima Dramé</strong>, <strong>Étienne Pardoux</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
In this work, we consider a continuous–time branching process with interaction where the birth and death rates are non linear functions of the population size. We prove that after a proper renormalization our model converges to a generalized continuous state branching process solution of the SDE \[\begin{aligned} Z_t^x=& x + \int _{0}^{t} f(Z_r^x) dr + \sqrt{2c} \int _{0}^{t} \int _{0}^{Z_{r}^x }W(dr,du) + \int _{0}^{t}\int _{0}^{1}\int _{0}^{Z_{r^-}^x}z \ \overline{M} (ds, dz, du)\\ &+ \int _{0}^{t}\int _{1}^{\infty }\int _{0}^{Z_{r^-}^x}z \ M(ds, dz, du), \end{aligned} \] where $W$ is a space-time white noise on $(0,\infty )^2$ and $\overline{M} (ds, dz, du)= M(ds, dz, du)- ds \mu (dz) du$, with $M$ being a Poisson random measure on $(0,\infty )^3$ independent of $W,$ with mean measure $ds\mu (dz)du$, where $(1\wedge z^2)\mu (dz)$ is a finite measure on $(0, \infty )$.
</p>projecteuclid.org/euclid.ecp/1539763346_20181123221011Fri, 23 Nov 2018 22:10 ESTSquared Bessel processes of positive and negative dimension embedded in Brownian local timeshttps://projecteuclid.org/euclid.ecp/1539763347<strong>Jim Pitman</strong>, <strong>Matthias Winkel</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
The Ray–Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \mu \ell $ of a reflecting Brownian motion $|B|$ by a multiple $\mu $ of its local time process $\ell $ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B$. This is done by decomposing the path of $B$ into its excursions above and below a family of continuous random levels determined by the Harrison–Shepp construction of skew Brownian motion as the strong solution of an SDE driven by $B$. This embedding connects to Brownian local times a framework of point processes of squared Bessel excursions of negative dimension and associated stable processes, recently introduced by Forman, Pal, Rizzolo and Winkel to set up interval partition evolutions that arise in their approach to the Aldous diffusion on a space of continuum trees.
</p>projecteuclid.org/euclid.ecp/1539763347_20181123221011Fri, 23 Nov 2018 22:10 ESTBiggins’ martingale convergence for branching Lévy processeshttps://projecteuclid.org/euclid.ecp/1540433049<strong>Jean Bertoin</strong>, <strong>Bastien Mallein</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
A branching Lévy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet $(\sigma ^2,a,\Lambda )$, where the branching Lévy measure $\Lambda $ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins’ theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet $(\sigma ^2,a,\Lambda )$ for additive martingales to have a non-degenerate limit.
</p>projecteuclid.org/euclid.ecp/1540433049_20181123221011Fri, 23 Nov 2018 22:10 ESTA stochastic model for the evolution of species with random fitnesshttps://projecteuclid.org/euclid.ecp/1543028983<strong>Daniela Bertacchi</strong>, <strong>Jüri Lember</strong>, <strong>Fabio Zucca</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
We generalize the evolution model introduced by Guiol, Machado and Schinazi (2010). In our model at odd times a random number $X$ of species is created. Each species is endowed with a random fitness with arbitrary distribution on $[0,1]$. At even times a random number $Y$ of species is removed, killing the species with lower fitness. We show that there is a critical fitness $f_c$ below which the number of species hits zero i.o. and above of which this number goes to infinity. We prove uniform convergence for the fitness distribution of surviving species and describe the phenomena which could not be observed in previous works with uniformly distributed fitness.
</p>projecteuclid.org/euclid.ecp/1543028983_20181123221011Fri, 23 Nov 2018 22:10 ESTSlowdown estimates for one-dimensional random walks in random environment with holding timeshttps://projecteuclid.org/euclid.ecp/1543028986<strong>Amir Dembo</strong>, <strong>Ryoki Fukushima</strong>, <strong>Naoki Kubota</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the probability that the random walk travels slower than its typical speed and determine its decay rate asymptotic.
</p>projecteuclid.org/euclid.ecp/1543028986_20181123221011Fri, 23 Nov 2018 22:10 EST