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 ESTSome properties for Itô processes driven by $G$-Brownian motionhttps://projecteuclid.org/euclid.ecp/1504922431<strong>Xinpeng Li</strong>, <strong>Falei Wang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 10 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we use partial differential equation (PDE) techniques and probabilistic approaches to study the lower capacity of the ball for the Itô process driven by $G$-Brownian motion ($G$-Itô process). In particular, the lower bound on the lower capacity of certain balls is obtained. As an application, we prove a strict comparison theorem in $G$-expectation framework.
</p>projecteuclid.org/euclid.ecp/1504922431_20170908220057Fri, 08 Sep 2017 22:00 EDTThe moving particle lemma for the exclusion process on a weighted graphhttps://projecteuclid.org/euclid.ecp/1506931447<strong>Joe P. Chen</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 13 pp..</p><p><strong>Abstract:</strong><br/>
We prove a version of the moving particle lemma for the exclusion process on any finite weighted graph, based on the octopus inequality of Caputo, Liggett, and Richthammer. In light of their proof of Aldous’ spectral gap conjecture, we conjecture that our moving particle lemma is optimal in general. Our result can be applied to graphs which lack translational invariance, including, but not limited to, fractal graphs. An application of our result is the proof of local ergodicity for the exclusion process on a class of weighted graphs, the details of which are reported in a follow-up paper [ arXiv:1705.10290 ].
</p>projecteuclid.org/euclid.ecp/1506931447_20171002040429Mon, 02 Oct 2017 04:04 EDTAsymptotic number of caterpillars of regularly varying $\Lambda $-coalescents that come down from infinityhttps://projecteuclid.org/euclid.ecp/1506931448<strong>Batı Şengül</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we look at the asymptotic number of $r$-caterpillars for $\Lambda $-coalescents which come down from infinity, under a regularly varying assumption. An $r$-caterpillar is a functional of the coalescent process started from $n$ individuals which, roughly speaking, is a block of the coalescent at some time, formed by one line of descend to which $r-1$ singletons have merged one by one. We show that the number of $r$-caterpillars, suitably scaled, converge to an explicit constant as the sample size $n$ goes to $\infty $.
</p>projecteuclid.org/euclid.ecp/1506931448_20171002040429Mon, 02 Oct 2017 04:04 EDTOn the multifractal local behavior of parabolic stochastic PDEshttps://projecteuclid.org/euclid.ecp/1506931449<strong>Jingyu Huang</strong>, <strong>Davar Khoshnevisan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 11 pp..</p><p><strong>Abstract:</strong><br/>
Consider the stochastic heat equation $\dot{u} =\frac 12 u''+\sigma (u)\xi $ on $(0\,,\infty )\times \mathbb{R} $ subject to $u(0)\equiv 1$, where $\sigma :\mathbb{R} \to \mathbb{R} $ is a Lipschitz (local) function that does not vanish at $1$, and $\xi $ denotes space-time white noise. It is well known that $u$ has continuous sample functions [22]; as a result, $\lim _{t\downarrow 0}u(t\,,x)= 1$ almost surely for every $x\in \mathbb{R} $.
The corresponding fluctuations are also known [14, 16, 20]: For every fixed $x\in \mathbb{R} $, $t\mapsto u(t\,,x)$ looks locally like a fixed multiple of fractional Brownian motion (fBm) with index $1/4$. In particular, an application of Fubini’s theorem implies that, on an $x$-set of full Lebesgue measure, the short-time behavior of the peaks of the random function $t\mapsto u(t\,,x)$ are governed by the law of the iterated logarithm for fBm, up to possibly a suitable normalization constant. By contrast, the main result of this paper claims that, on an $x$-set of full Hausdorff dimension, the short-time peaks of $t\mapsto u(t\,,x)$ follow a non-iterated logarithm law, and that those peaks contain a rich multifractal structure a.s.
Large-time variations of these results were predicted in the physics literature a number of years ago and proved very recently in [10, 11]. To the best of our knowledge, the short-time results of the present paper are observed here for the first time.
</p>projecteuclid.org/euclid.ecp/1506931449_20171002040429Mon, 02 Oct 2017 04:04 EDTOn the threshold of spread-out voter model percolationhttps://projecteuclid.org/euclid.ecp/1507255233<strong>Balázs Ráth</strong>, <strong>Daniel Valesin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z} ^d$ has state (or ‘opinion’) 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal) stationary measures of this model is given by a family $\mu _{\alpha , R}$, where $\alpha \in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0’s and 1’s in which the density of 1’s is $\alpha $ and the correlation weakens as $R$ becomes larger. We study these configurations from the point of view of nearest neighbor site percolation on $\mathbb{Z} ^d$, focusing on asymptotics as $R \to \infty $. In [RV15], we have shown that, if $R$ is large, there is a critical value $\alpha _c(R)$ such that there is percolation if $\alpha > \alpha _c(R)$ and no percolation if $\alpha < \alpha _c(R)$. Here we prove that, as $R \to \infty $, $\alpha _c(R)$ converges to the critical probability for Bernoulli site percolation on $\mathbb{Z} ^d$. Our proof relies on a new upper bound on the joint occurrence of events under $\mu _{\alpha ,R}$ which is of independent interest.
</p>projecteuclid.org/euclid.ecp/1507255233_20171005220134Thu, 05 Oct 2017 22:01 EDTA heat flow approach to the Godbillon-Vey classhttps://projecteuclid.org/euclid.ecp/1483585771<strong>Diego S. Ledesma</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 6 pp..</p><p><strong>Abstract:</strong><br/>
We give a heat flow derivation for the Godbillon Vey class. In particular we prove that if $(M,g)$ is a compact Riemannian manifold with a codimension 1 foliation $\mathcal{F} $, defined by an integrable 1-form $\omega $ such that $||\omega ||=1$, then the Godbillon-Vey class can be written as $[-\mathcal{A} \omega \wedge d\omega ]_{dR}$ for an operator $\mathcal{A} :\Omega ^*(M)\rightarrow \Omega ^*(M)$ induced by the heat flow.
</p>projecteuclid.org/euclid.ecp/1483585771_20171009040719Mon, 09 Oct 2017 04:07 EDTInequalities for the Gaussian measure of convex setshttps://projecteuclid.org/euclid.ecp/1507536392<strong>Michael R. Tehranchi</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 7 pp..</p><p><strong>Abstract:</strong><br/>
This note presents families of inequalities for the Gaussian measure of convex sets which extend the recently proven Gaussian correlation inequality in various directions.
</p>projecteuclid.org/euclid.ecp/1507536392_20171009040719Mon, 09 Oct 2017 04:07 EDTApplication of stochastic flows to the sticky Brownian motion equationhttps://projecteuclid.org/euclid.ecp/1483585772<strong>Hatem Hajri</strong>, <strong>Mine Caglar</strong>, <strong>Marc Arnaudon</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 10 pp..</p><p><strong>Abstract:</strong><br/>
We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation.
</p>projecteuclid.org/euclid.ecp/1483585772_20171012220417Thu, 12 Oct 2017 22:04 EDTRecurrence and transience properties of multi-dimensional diffusion processes in selfsimilar and semi-selfsimilar random environmentshttps://projecteuclid.org/euclid.ecp/1483585773<strong>Seiichiro Kusuoka</strong>, <strong>Hiroshi Takahashi</strong>, <strong>Yozo Tamura</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 11 pp..</p><p><strong>Abstract:</strong><br/>
We consider $d$-dimensional diffusion processes in multi-parameter random environments which are given by values at different $d$ points of one-dimensional $\alpha $-stable or $(r, \alpha )$-semi-stable Lévy processes. From the model, we derive some conditions of random environments that imply the dichotomy of recurrence and transience for the $d$-dimensional diffusion processes. The limiting behavior is quite different from that of a $d$-dimensional standard Brownian motion. We also consider the direct product of a one-dimensional diffusion process in a reflected non-positive Brownian environment and a one-dimensional standard Brownian motion. For the two-dimensional diffusion process, we show the transience property for almost all reflected Brownian environments.
</p>projecteuclid.org/euclid.ecp/1483585773_20171012220417Thu, 12 Oct 2017 22:04 EDTA Markov chain representation of the normalized Perron–Frobenius eigenvectorhttps://projecteuclid.org/euclid.ecp/1507860209<strong>Raphaël Cerf</strong>, <strong>Joseba Dalmau</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 6 pp..</p><p><strong>Abstract:</strong><br/>
We consider the problem of finding the Perron–Frobenius eigenvector of a primitive matrix. Dividing each of the rows of the matrix by the sum of the elements in the row, the resulting new matrix is stochastic. We give a formula for the normalized Perron–Frobenius eigenvector of the original matrix, in terms of a realization of the Markov chain defined by the associated stochastic matrix. This formula is a generalization of the classical formula for the invariant probability measure of a Markov chain.
</p>projecteuclid.org/euclid.ecp/1507860209_20171012220417Thu, 12 Oct 2017 22:04 EDTStochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficientshttps://projecteuclid.org/euclid.ecp/1507860210<strong>Eduardo Abi Jaber</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 15 pp..</p><p><strong>Abstract:</strong><br/>
We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of $\mathbb{R} ^d$ with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.
</p>projecteuclid.org/euclid.ecp/1507860210_20171012220417Thu, 12 Oct 2017 22:04 EDTOn the sub-Gaussianity of the Beta and Dirichlet distributionshttps://projecteuclid.org/euclid.ecp/1507860211<strong>Olivier Marchal</strong>, <strong>Julyan Arbel</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 14 pp..</p><p><strong>Abstract:</strong><br/>
We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta moment-generating function known as the confluent hypergeometric function. As a consequence, we derive the optimal proxy variance for the Dirichlet distribution, which is apparently a novel result. We also provide a new proof of the optimal proxy variance for the Bernoulli distribution, and discuss in this context the proxy variance relation to log-Sobolev inequalities and transport inequalities.
</p>projecteuclid.org/euclid.ecp/1507860211_20171012220417Thu, 12 Oct 2017 22:04 EDTDonsker-type theorems for correlated geometric fractional Brownian motions and related processeshttps://projecteuclid.org/euclid.ecp/1507860212<strong>Peter Parczewski</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 13 pp..</p><p><strong>Abstract:</strong><br/>
We prove a Donsker-type theorem for vector processes of functionals of correlated Wiener integrals. This includes the case of correlated geometric fractional Brownian motions of arbitrary Hurst parameters in $(0,1)$ driven by the same Brownian motion. Starting from a Donsker-type approximation of Wiener integrals of Volterra type by disturbed binary random walks, the continuous and discrete Wiener chaos representation in terms of Wick calculus is effective. The main result is the compatibility of these continuous and discrete stochastic calculi via these multivariate limit theorems.
</p>projecteuclid.org/euclid.ecp/1507860212_20171012220417Thu, 12 Oct 2017 22:04 EDTConvergence rates of the random scan Gibbs sampler under the Dobrushin’s uniqueness conditionhttps://projecteuclid.org/euclid.ecp/1507860213<strong>Neng-Yi Wang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 7 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, under the Dobrushin’s uniqueness condition, we obtain explicit estimates of the geometrical convergence rate for the random scan Gibbs sampler in the Wasserstein metric.
</p>projecteuclid.org/euclid.ecp/1507860213_20171012220417Thu, 12 Oct 2017 22:04 EDTRecurrence of multiply-ended planar triangulationshttps://projecteuclid.org/euclid.ecp/1483671681<strong>Ori Gurel-Gurevich</strong>, <strong>Asaf Nachmias</strong>, <strong>Juan Souto</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 6 pp..</p><p><strong>Abstract:</strong><br/>
In this note we show that a bounded degree planar triangulation is recurrent if and only if the set of accumulation points of some/any circle packing of it is polar (that is, planar Brownian motion avoids it with probability $1$). This generalizes a theorem of He and Schramm [6] who proved it when the set of accumulation points is either empty or a Jordan curve, in which case the graph has one end. We also show that this statement holds for any straight-line embedding with angles uniformly bounded away from $0$.
</p>projecteuclid.org/euclid.ecp/1483671681_20171017220214Tue, 17 Oct 2017 22:02 EDTUniversal large deviations for Kac polynomialshttps://projecteuclid.org/euclid.ecp/1483952415<strong>Raphaël Butez</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 10 pp..</p><p><strong>Abstract:</strong><br/>
We prove the universality of the large deviations principle for the empirical measures of zeros of random polynomials whose coefficients are i.i.d. random variables possessing a density with respect to the Lebesgue measure on $\mathbb{C} $, $\mathbb{R} $ or $\mathbb{R} ^+$, under the assumption that the density does not vanish too fast at zero and decays at least as $\exp -|x|^{\rho }$, $\rho >0$, at infinity.
</p>projecteuclid.org/euclid.ecp/1483952415_20171017220214Tue, 17 Oct 2017 22:02 EDTOn recurrence and transience of multivariate near-critical stochastic processeshttps://projecteuclid.org/euclid.ecp/1484190065<strong>Götz Kersting</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
We obtain complementary recurrence/transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ \xi _{n+1}$. Here $M$ denotes a primitive matrix having Perron-Frobenius eigenvalue 1, and $g$ denotes some function. The conditional expectation and variance of the noise $(\xi _{n+1})_{n \ge 0}$ are such that $X$ obeys a weak form of the Markov property. The results generalize criteria for the 1-dimensional case in [5].
</p>projecteuclid.org/euclid.ecp/1484190065_20171017220214Tue, 17 Oct 2017 22:02 EDTYet another condition for absence of collisions for competing Brownian particleshttps://projecteuclid.org/euclid.ecp/1484363135<strong>Tomoyuki Ichiba</strong>, <strong>Andrey Sarantsev</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 7 pp..</p><p><strong>Abstract:</strong><br/>
Consider a finite system of rank-based competing Brownian particles, where the drift and diffusion of each particle depend only on its current rank relative to other particles. We present a simple sufficient condition for absence of multiple collisions of a given order, continuing the earlier work by Bruggeman and Sarantsev (2015). Unlike in that paper, this new condition works even for infinite systems.
</p>projecteuclid.org/euclid.ecp/1484363135_20171017220214Tue, 17 Oct 2017 22:02 EDTStable limit theorem for $U$-statistic processes indexed by a random walkhttps://projecteuclid.org/euclid.ecp/1484363136<strong>Brice Franke</strong>, <strong>Françoise Pène</strong>, <strong>Martin Wendler</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
Let $(S_n)_{n\in \mathbb{N} }$ be a $\mathbb{Z} $-valued random walk with increments from the domain of attraction of some $\alpha $-stable law and let $(\xi (i))_{i\in \mathbb{Z} }$ be a sequence of iid random variables. We want to investigate $U$-statistics indexed by the random walk $S_n$, that is $U_n:=\sum _{1\leq i<j\leq n}h(\xi (S_i),\xi (S_j))$ for some symmetric bivariate function $h$. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the $U$-statistic $U_n$.
</p>projecteuclid.org/euclid.ecp/1484363136_20171017220214Tue, 17 Oct 2017 22:02 EDTNecessary and sufficient conditions for the $r$-excessive local martingales to be martingaleshttps://projecteuclid.org/euclid.ecp/1485421233<strong>Mikhail Urusov</strong>, <strong>Mihail Zervos</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 6 pp..</p><p><strong>Abstract:</strong><br/>
We consider the decreasing and the increasing $r$-excessive functions $\varphi _r$ and $\psi _r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints $\alpha < \beta $. We prove that the $r$-excessive local martingale $\bigl ( e^{-r (t \wedge T_\alpha )} \varphi _r (X_{t \wedge T_\alpha }) \bigr )$ $\bigl ($resp., $\bigl ( e^{-r (t \wedge T_\beta )} \psi _r (X_{t \wedge T_\beta }) \bigr ) \bigr )$ is a strict local martingale if the boundary point $\alpha $ (resp., $\beta $) is inaccessible and entrance, and a martingale otherwise.
</p>projecteuclid.org/euclid.ecp/1485421233_20171017220214Tue, 17 Oct 2017 22:02 EDTProduct space for two processes with independent increments under nonlinear expectationshttps://projecteuclid.org/euclid.ecp/1485421234<strong>Qiang Gao</strong>, <strong>Mingshang Hu</strong>, <strong>Xiaojun Ji</strong>, <strong>Guomin Liu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the product space for two processes with independent increments under nonlinear expectations. By introducing a discretization method, we construct a nonlinear expectation under which the given two processes can be seen as a new process with independent increments.
</p>projecteuclid.org/euclid.ecp/1485421234_20171017220214Tue, 17 Oct 2017 22:02 EDTAn inequality for the heat kernel on an Abelian Cayley graphhttps://projecteuclid.org/euclid.ecp/1508292095<strong>Thomas McMurray Price</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 8 pp..</p><p><strong>Abstract:</strong><br/>
We demonstrate a relationship between the heat kernel on a finite weighted Abelian Cayley graph and Gaussian functions on lattices. This can be used to prove a new inequality for the heat kernel on such a graph: when $t \leq t'$, \[ \frac{H_t(u, v)} {H_t(u,u)} \leq \frac{H_{t'}(u, v)} {H_{t'}(u,u)}. \] This was an open problem posed by Regev and Shinkar.
</p>projecteuclid.org/euclid.ecp/1508292095_20171017220214Tue, 17 Oct 2017 22:02 EDTMultidimensional quadratic BSDEs with separated generatorshttps://projecteuclid.org/euclid.ecp/1508292096<strong>Asgar Jamneshan</strong>, <strong>Michael Kupper</strong>, <strong>Peng Luo</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 10 pp..</p><p><strong>Abstract:</strong><br/>
We consider multidimensional quadratic BSDEs with bounded and unbounded terminal conditions. We provide sufficient conditions which guarantee existence and uniqueness of solutions. In particular, these conditions are satisfied if the terminal condition or the dependence in the system are small enough.
</p>projecteuclid.org/euclid.ecp/1508292096_20171017220214Tue, 17 Oct 2017 22:02 EDTNote on a one-dimensional system of annihilating particleshttps://projecteuclid.org/euclid.ecp/1508292097<strong>Vladas Sidoravicius</strong>, <strong>Laurent Tournier</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 9 pp..</p><p><strong>Abstract:</strong><br/>
We consider a system of annihilating particles where particles start from the points of a Poisson process on either the full-line or positive half-line and move at constant i.i.d. speeds until collision. When two particles collide, they annihilate. We assume the law of speeds to be symmetric. We prove almost sure annihilation of positive-speed particles started from the positive half-line, and existence of a regime of survival of zero-speed particles on the full-line in the case when speeds can only take 3 values. We also state open questions.
</p>projecteuclid.org/euclid.ecp/1508292097_20171017220214Tue, 17 Oct 2017 22:02 EDTThe set of connective constants of Cayley graphs contains a Cantor spacehttps://projecteuclid.org/euclid.ecp/1485507643<strong>Sébastien Martineau</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 4 pp..</p><p><strong>Abstract:</strong><br/>
The connective constant of a transitive graph is the exponential growth rate of its number of self-avoiding walks. We prove that the set of connective constants of the so-called Cayley graphs contains a Cantor set. In particular, this set has the cardinality of the continuum.
</p>projecteuclid.org/euclid.ecp/1485507643_20171024220327Tue, 24 Oct 2017 22:03 EDTIndicable groups and $p_c<1$https://projecteuclid.org/euclid.ecp/1485831618<strong>Aran Raoufi</strong>, <strong>Ariel Yadin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 10 pp..</p><p><strong>Abstract:</strong><br/> A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of $\mathbb{Z} $ has a non-trivial percolation phase. Our main results prove this conjecture for certain groups, and in particular prove that any group with a non-trivial homomorphism into the additive group of real numbers satisfies the conjecture. We use this to reduce the conjecture to the case of hereditary just-infinite groups. The novelty here is mainly in the methods used, combining the methods of EIT and evolving sets, and using the algebraic properties of the group to apply these methods. </p>projecteuclid.org/euclid.ecp/1485831618_20171024220327Tue, 24 Oct 2017 22:03 EDTFirst passage percolation on a hyperbolic graph admits bi-infinite geodesicshttps://projecteuclid.org/euclid.ecp/1487062816<strong>Itai Benjamini</strong>, <strong>Romain Tessera</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 8 pp..</p><p><strong>Abstract:</strong><br/>
Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg ([14]) is whether there exists a bi-infinite geodesic in first passage percolation on the euclidean lattice of dimension at least 2. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite expectation of the random lengths, we prove that if a graph $X$ has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost surely, there exists a bi-infinite geodesic in first passage percolation on $X$.
</p>projecteuclid.org/euclid.ecp/1487062816_20171024220327Tue, 24 Oct 2017 22:03 EDTErratum: Optimal linear drift for the speed of convergence of an hypoelliptic diffusionhttps://projecteuclid.org/euclid.ecp/1487062817<strong>Arnaud Guillin</strong>, <strong>Pierre Monmarché</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 2 pp..</p><p><strong>Abstract:</strong><br/>
Erratum for Optimal linear drift for the speed of convergence of an hypoelliptic diffusion , A. Guillin, and P. Monmarché, Electron. Commun. Probab. 21 (2016), paper no. 74, 14 pp. doi:10.1214/16-ECP25.
</p>projecteuclid.org/euclid.ecp/1487062817_20171024220327Tue, 24 Oct 2017 22:03 EDTSelf-averaging sequences which fail to convergehttps://projecteuclid.org/euclid.ecp/1487386904<strong>Eric Cator</strong>, <strong>Henk Don</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is mainly based on terms around a fixed fraction of $n$. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.
</p>projecteuclid.org/euclid.ecp/1487386904_20171024220327Tue, 24 Oct 2017 22:03 EDTSome connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisetshttps://projecteuclid.org/euclid.ecp/1487386905<strong>Ross G. Pinsky</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 9 pp..</p><p><strong>Abstract:</strong><br/>
We present a new proof of a fundamental result concerning cycles of random permutations which gives some intuition for the connection between Touchard polynomials and the Poisson distribution. We also introduce a rather novel permutation statistic and study its distribution. This quantity, indexed by $m$, is the number of sets of size $m$ fixed by the permutation. This leads to a new and simpler derivation of the exponential generating function for the number of covers of certain multisets.
</p>projecteuclid.org/euclid.ecp/1487386905_20171024220327Tue, 24 Oct 2017 22:03 EDTConvergence of complex martingales in the branching random walk: the boundaryhttps://projecteuclid.org/euclid.ecp/1488337251<strong>Konrad Kolesko</strong>, <strong>Matthias Meiners</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 14 pp..</p><p><strong>Abstract:</strong><br/>
Biggins [Uniform convergence of martingales in the branching random walk. Ann. Probab. , 20(1):137–151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\lambda $ from an open set $\Lambda \subseteq \mathbb{C} ^d$. We investigate the martingales corresponding to parameters from the boundary $\partial \Lambda $ of $\Lambda $. The boundary can be decomposed into several parts. We demonstrate by means of an example that there may be a part of the boundary, on which the martingales do not exist. Where the martingales exist, they may diverge, vanish in the limit or converge to a non-degenerate limit. We provide mild sufficient conditions for each of these three types of limiting behaviors to occur. The arguments that give convergence to a non-degenerate limit also apply in $\Lambda $ and require weaker moment assumptions than the ones used by Biggins.
</p>projecteuclid.org/euclid.ecp/1488337251_20171024220327Tue, 24 Oct 2017 22:03 EDTRemarks on spectral gaps on the Riemannian path spacehttps://projecteuclid.org/euclid.ecp/1489457081<strong>Shizan Fang</strong>, <strong>Bo Wu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 13 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces.
</p>projecteuclid.org/euclid.ecp/1489457081_20171024220327Tue, 24 Oct 2017 22:03 EDTGalton-Watson probability contractionhttps://projecteuclid.org/euclid.ecp/1489651213<strong>Moumanti Podder</strong>, <strong>Joel Spencer</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 16 pp..</p><p><strong>Abstract:</strong><br/>
We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with $Poisson(c)$ offspring distribution. Fixing a positive integer $k$, we exploit the $k$-move Ehrenfeucht game on rooted trees for this purpose. Let $\Sigma $, indexed by $1 \leq j \leq m$, denote the finite set of equivalence classes arising out of this game, and $D$ the set of all probability distributions over $\Sigma $. Let $x_{j}(c)$ denote the true probability of the class $j \in \Sigma $ under $Poisson(c)$ regime, and $\vec{x} (c)$ the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function $\Gamma $, and a map $\Psi = \Psi _{c}: D \rightarrow D$ such that $\vec{x} (c)$ is a fixed point of $\Psi _{c}$, and starting with any distribution $\vec{x} \in D$, we converge to this fixed point via $\Psi $ because it is a contraction. We show this both for $c \leq 1$ and $c > 1$, though the techniques for these two ranges are quite different.
</p>projecteuclid.org/euclid.ecp/1489651213_20171024220327Tue, 24 Oct 2017 22:03 EDTThe existence phase transition for two Poisson random fractal modelshttps://projecteuclid.org/euclid.ecp/1490839345<strong>Erik I. Broman</strong>, <strong>Johan Jonasson</strong>, <strong>Johan Tykesson</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 8 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we study the existence phase transition of the random fractal ball model and the random fractal box model. We show that both of these are in the empty phase at the critical point of this phase transition.
</p>projecteuclid.org/euclid.ecp/1490839345_20171024220327Tue, 24 Oct 2017 22:03 EDTInformation loss on Gaussian Volterra processhttps://projecteuclid.org/euclid.ecp/1508896983<strong>Arturo Valdivia</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 5 pp..</p><p><strong>Abstract:</strong><br/>
Gaussian Volterra processes are processes of the form $(X_{t}:=\int _{\mathbf{T} }k(t,s)\mathrm{d} W_{s})_{t\in \mathbf{T} }$ where $(W_{t})_{t\in \mathbf{T} }$ is Brownian motion, and $k$ is a deterministic Volterra kernel. On integrating the kernel $k$ an information loss may occur, in the sense that the filtration of the Volterra process needs to be enlarged in order to recover the filtration of the driving Brownian motion. In this note we describe such enlargement of filtrations in terms of the Volterra kernel. For kernels of the form $k(t,s)=k(t-s)$ we provide a simple criterion to ensure that the aforementioned filtrations coincide.
</p>projecteuclid.org/euclid.ecp/1508896983_20171024220327Tue, 24 Oct 2017 22:03 EDTImproved bounds for the mixing time of the random-to-random shufflehttps://projecteuclid.org/euclid.ecp/1491271396<strong>Chuan Qin</strong>, <strong>Ben Morris</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 7 pp..</p><p><strong>Abstract:</strong><br/>
We prove an upper bound of $1.5321 n \log n$ for the mixing time of the random-to-random insertion shuffle, improving on the best known upper bound of $2 n \log n$. Our proof is based on the analysis of a non-Markovian coupling.
</p>projecteuclid.org/euclid.ecp/1491271396_20171115040041Wed, 15 Nov 2017 04:00 ESTNote on A. Barbour’s paper on Stein’s method for diffusion approximationshttps://projecteuclid.org/euclid.ecp/1492221618<strong>Mikołaj J. Kasprzak</strong>, <strong>Andrew B. Duncan</strong>, <strong>Sebastian J. Vollmer</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 8 pp..</p><p><strong>Abstract:</strong><br/>
In [2] foundations for diffusion approximation via Stein’s method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein’s method (see, for example, its use in [1] or [7]). A semigroup argument is used in [2] to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in [2], the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on $D[0,1]$ growing slower than a cubic, equipped with an appropriate norm. We also provide a proof of the exact formulation of the solution to the Stein equation of interest, which does not require the aforementioned strong continuity. This shows that the main results of [2] hold true.
</p>projecteuclid.org/euclid.ecp/1492221618_20171115040041Wed, 15 Nov 2017 04:00 ESTStein type characterization for $G$-normal distributionshttps://projecteuclid.org/euclid.ecp/1492588926<strong>Mingshang Hu</strong>, <strong>Shige Peng</strong>, <strong>Yongsheng Song</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this article, we provide a Stein type characterization for $G$-normal distributions: Let $\mathcal{N} [\varphi ]=\sup _{\mu \in \Theta }\mu [\varphi ],\ \varphi \in C_{b,Lip}(\mathbb{R} ),$ be a sublinear expectation. $\mathcal{N} $ is $G$-normal if and only if for any $\varphi \in C_b^2(\mathbb{R} )$, we have \[ \int _\mathbb{R} [\frac{x} {2}\varphi '(x)-G(\varphi ''(x))]\mu ^\varphi (dx)=0, \] where $\mu ^\varphi $ is a realization of $\varphi $ associated with $\mathcal{N} $, i.e., $\mu ^\varphi \in \Theta $ and $\mu ^\varphi [\varphi ]=\mathcal{N} [\varphi ]$.
</p>projecteuclid.org/euclid.ecp/1492588926_20171115040041Wed, 15 Nov 2017 04:00 ESTTwo observations on the capacity of the range of simple random walks on $\mathbb{Z} ^3$ and $\mathbb{Z} ^4$https://projecteuclid.org/euclid.ecp/1493777159<strong>Yinshan Chang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 9 pp..</p><p><strong>Abstract:</strong><br/>
We prove a weak law of large numbers for the capacity of the range of simple random walks on $\mathbb{Z} ^{4}$. On $\mathbb{Z} ^{3}$, we show that the capacity, properly scaled, converges in distribution towards the corresponding quantity for three dimensional Brownian motion. The paper answers two of the three open questions raised by Asselah, Schapira and Sousi in [2, Section 6].
</p>projecteuclid.org/euclid.ecp/1493777159_20171115040041Wed, 15 Nov 2017 04:00 ESTKesten’s incipient infinite cluster and quasi-multiplicativity of crossing probabilitieshttps://projecteuclid.org/euclid.ecp/1494036081<strong>Deepan Basu</strong>, <strong>Artem Sapozhnikov</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph $G$. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of Kesten’s incipient infinite cluster. We show that our assumptions are satisfied if $G$ is a slab $\mathbb{Z} ^2\times \{0,\ldots ,k\}^{d-2}$ ($d\geq 2$, $k\geq 0$). We also argue that the quasi-multiplicativity assumption should hold for $G=\mathbb{Z} ^d$ when $d<6$, but not when $d>6$.
</p>projecteuclid.org/euclid.ecp/1494036081_20171115040041Wed, 15 Nov 2017 04:00 ESTAn elementary approach to Gaussian multiplicative chaoshttps://projecteuclid.org/euclid.ecp/1494554429<strong>Nathanaël Berestycki</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase $(\gamma < \sqrt{2d} )$ and that the limit is universal (i.e., the limiting measure is independent of the regularisation of the underlying field).
</p>projecteuclid.org/euclid.ecp/1494554429_20171115040041Wed, 15 Nov 2017 04:00 ESTAbout the constants in the Fuk-Nagaev inequalitieshttps://projecteuclid.org/euclid.ecp/1494921750<strong>Emmanuel Rio</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we give efficient constants in the Fuk-Nagaev inequalities. Next we derive new upper bounds on the weak norms of martingales from our Fuk-Nagaev type inequality.
</p>projecteuclid.org/euclid.ecp/1494921750_20171115040041Wed, 15 Nov 2017 04:00 ESTSurvival asymptotics for branching random walks in IID environmentshttps://projecteuclid.org/euclid.ecp/1495677741<strong>János Engländer</strong>, <strong>Yuval Peres</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/> We first study a model, introduced recently in [4], of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no ‘obstacle’ placed there. The obstacles appear at each site with probability $p\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\mathbb P_p$-measure, as $n\to \infty $, $P^{\omega }(S_n)\sim 2/(qn)$ in the critical case, while this probability is asymptotically stretched exponential in the subcritical case. Hence, the model exhibits ‘self-averaging’ in the critical case but not in the subcritical one. I.e., in the first case, the asymptotic tail behavior is the same as in a ‘toy model’ where space is removed, while in the second, the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. A spine decomposition of the branching process along with known results on random walks are utilized. </p>projecteuclid.org/euclid.ecp/1495677741_20171115040041Wed, 15 Nov 2017 04:00 ESTThe frog model with drift on $\mathbb{R} $https://projecteuclid.org/euclid.ecp/1495764227<strong>Joshua Rosenberg</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 14 pp..</p><p><strong>Abstract:</strong><br/>
Consider a Poisson process on $\mathbb{R} $ with intensity $f$ where $0 \leq f(x)<\infty $ for ${x}\geq 0$ and ${f(x)}=0$ for $x<0$. The “points” of the process represent sleeping frogs. In addition, there is one active frog initially located at the origin. At time ${t}=0$ this frog begins performing Brownian motion with leftward drift $\lambda $ (i.e. its motion is a random process of the form ${B}_{t}-\lambda{t} $). Any time an active frog arrives at a point where a sleeping frog is residing, the sleeping frog becomes active and begins performing Brownian motion with leftward drift $\lambda $, independently of the motion of all of the other active frogs. This paper establishes sharp conditions on the intensity function $f$ that determine whether the model is transient (meaning the probability that infinitely many frogs return to the origin is 0), or non-transient (meaning this probability is greater than 0). A discrete model with $\text{Poiss} (f(n))$ sleeping frogs at positive integer points (and where activated frogs perform biased random walks on $\mathbb{Z} $) is also examined. In this case as well, we obtain a similar sharp condition on $f$ corresponding to transience of the model.
</p>projecteuclid.org/euclid.ecp/1495764227_20171115040041Wed, 15 Nov 2017 04:00 ESTPiecewise constant local martingales with bounded numbers of jumpshttps://projecteuclid.org/euclid.ecp/1496282536<strong>Johannes Ruf</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 5 pp..</p><p><strong>Abstract:</strong><br/>
A piecewise constant local martingale $M$ with boundedly many jumps is a uniformly integrable martingale if and only if $M_\infty ^-$ is integrable.
</p>projecteuclid.org/euclid.ecp/1496282536_20171115040041Wed, 15 Nov 2017 04:00 ESTMarčenko-Pastur law for Kendall’s tauhttps://projecteuclid.org/euclid.ecp/1496455233<strong>Afonso S. Bandeira</strong>, <strong>Asad Lodhia</strong>, <strong>Philippe Rigollet</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 7 pp..</p><p><strong>Abstract:</strong><br/>
We prove that Kendall’s Rank correlation matrix converges to the Marčenko Pastur law, under the assumption that observations are i.i.d random vectors $X_1, \ldots , X_n$ with components that are independent and absolutely continuous with respect to the Lebesgue measure. This is the first result on the empirical spectral distribution of a multivariate $U$-statistic.
</p>projecteuclid.org/euclid.ecp/1496455233_20171115040041Wed, 15 Nov 2017 04:00 ESTSecond order behavior of the block counting process of beta coalescentshttps://projecteuclid.org/euclid.ecp/1510736418<strong>Yier Lin</strong>, <strong>Bastien Mallein</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 8 pp..</p><p><strong>Abstract:</strong><br/>
The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of blocs at small times in the Beta coalescent. Berestycki, Berestycki and Schweinsberg [2] proved a law of large numbers for this quantity. Recently, Limic and Talarczyk [9] proved that a functional central limit theorem holds as well. We give here a simple proof for an unidimensional version of this result, using a coupling between Beta coalescents and continuous-time branching processes.
</p>projecteuclid.org/euclid.ecp/1510736418_20171115040041Wed, 15 Nov 2017 04:00 ESTA Cramér type moderate deviation theorem for the critical Curie-Weiss modelhttps://projecteuclid.org/euclid.ecp/1510736419<strong>Hao Can Can</strong>, <strong>Viet-Hung Pham</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
Limit theorems for the magnetization of Curie-Weiss model have been studied extensively by Ellis and Newman. To refine these results, Chen, Fang and Shao prove Cramér type moderate deviation theorems for non-critical cases by using Stein method. In this paper, we consider the same question for the remaining case - the critical Curie-Weiss model. By direct and simple arguments based on Laplace method, we provide an explicit formula of the error and deduce a Cramér type result.
</p>projecteuclid.org/euclid.ecp/1510736419_20171115040041Wed, 15 Nov 2017 04:00 ESTUniform convergence to the $Q$-processhttps://projecteuclid.org/euclid.ecp/1497319609<strong>Nicolas Champagnat</strong>, <strong>Denis Villemonais</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 7 pp..</p><p><strong>Abstract:</strong><br/>
The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its $Q$-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a conditioned process converges uniformly to a conservative Markov process which is itself ergodic, then it admits a unique quasi-stationary distribution and converges toward it exponentially fast, uniformly in its initial distribution. As an application, we provide a conditional ergodic theorem.
</p>projecteuclid.org/euclid.ecp/1497319609_20171115221556Wed, 15 Nov 2017 22:15 ESTA note on Malliavin smoothness on the Lévy spacehttps://projecteuclid.org/euclid.ecp/1498010647<strong>Eija Laukkarinen</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider Malliavin calculus based on the Itô chaos decomposition of square integrable random variables on the Lévy space. We show that when a random variable satisfies a certain measurability condition, its differentiability and fractional differentiability can be determined by weighted Lebesgue spaces. The measurability condition is satisfied for all random variables if the underlying Lévy process is a compound Poisson process on a finite time interval.
</p>projecteuclid.org/euclid.ecp/1498010647_20171115221556Wed, 15 Nov 2017 22:15 ESTOn the relaxation rate of short chains of rotors interacting with Langevin thermostatshttps://projecteuclid.org/euclid.ecp/1498010648<strong>Noé Cuneo</strong>, <strong>Christophe Poquet</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 8 pp..</p><p><strong>Abstract:</strong><br/>
In this short note, we consider a system of two rotors, one of which interacts with a Langevin heat bath. We show that the system relaxes to its invariant measure (steady state) no faster than a stretched exponential $\exp (-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by the present authors and J.-P. Eckmann for short chains of rotors is optimal.
</p>projecteuclid.org/euclid.ecp/1498010648_20171115221556Wed, 15 Nov 2017 22:15 ESTA note on continuous-time stochastic approximation in infinite dimensionshttps://projecteuclid.org/euclid.ecp/1498204833<strong>Jan Seidler</strong>, <strong>František Žák</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 13 pp..</p><p><strong>Abstract:</strong><br/>
We find sufficient conditions for convergence of a continuous-time Robbins-Monro type stochastic approximation procedure in infinite dimensional Hilbert spaces in terms of Lyapunova functions, the variational approach to stochastic partial differential equations being used as the main tool.
</p>projecteuclid.org/euclid.ecp/1498204833_20171115221556Wed, 15 Nov 2017 22:15 ESTLarge deviations for biorthogonal ensembles and variational formulation for the Dykema-Haagerup distributionhttps://projecteuclid.org/euclid.ecp/1499068820<strong>Raphaël Butez</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 11 pp..</p><p><strong>Abstract:</strong><br/>
This note provides a large deviation principle for a class of biorthogonal ensembles. We extend the results of Eichelsbacher, Sommerauer and Stolz to a more general type of interactions. In particular, our result covers the case of the singular values of lower triangular random matrices with independent entries introduced by Cheliotis and implies a variational formulation for the Dykema–Haagerup distribution.
</p>projecteuclid.org/euclid.ecp/1499068820_20171115221556Wed, 15 Nov 2017 22:15 ESTOn the Semi-classical Brownian Bridge Measurehttps://projecteuclid.org/euclid.ecp/1501833630<strong>Xue-Mei Li</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 15 pp..</p><p><strong>Abstract:</strong><br/>
We prove an integration by parts formula for the probability measure on the pinned path space induced by the Semi-classical Riemmanian Brownian Bridge, over a manifold with a pole, followed by a discussion on its equivalence with the Brownian Bridge measure.
</p>projecteuclid.org/euclid.ecp/1501833630_20171115221556Wed, 15 Nov 2017 22:15 ESTA functional limit theorem for excited random walkshttps://projecteuclid.org/euclid.ecp/1502244193<strong>Andrey Pilipenko</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 9 pp..</p><p><strong>Abstract:</strong><br/>
We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.
</p>projecteuclid.org/euclid.ecp/1502244193_20171115221556Wed, 15 Nov 2017 22:15 ESTOn stochastic heat equation with measure initial datahttps://projecteuclid.org/euclid.ecp/1502416901<strong>Jingyu Huang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 6 pp..</p><p><strong>Abstract:</strong><br/>
The aim of this short note is to obtain the existence, uniqueness and moment upper bounds of the solution to a stochastic heat equation with measure initial data, without using the iteration method in [1, 2, 3].
</p>projecteuclid.org/euclid.ecp/1502416901_20171115221556Wed, 15 Nov 2017 22:15 ESTOn the real spectrum of a product of Gaussian matriceshttps://projecteuclid.org/euclid.ecp/1502416902<strong>Nick Simm</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 11 pp..</p><p><strong>Abstract:</strong><br/>
Let $X_{m} = G_{1}\ldots G_{m}$ denote the product of $m$ independent random matrices of size $N \times N$, with each matrix in the product consisting of independent standard Gaussian variables. Denoting by $N_{\mathbb{R} }(m)$ the total number of real eigenvalues of $X_{m}$, we show that for $m$ fixed \[ \mathbb{E} (N_{\mathbb{R} }(m)) = \sqrt{\frac {2Nm}{\pi }} +O(\log (N)), \qquad N \to \infty . \] This generalizes a well-known result of Edelman et al. [10] to all $m>1$. Furthermore, we show that the normalized global density of real eigenvalues converges weakly in expectation to the density of the random variable $|U|^{m}B$ where $U$ is uniform on $[-1,1]$ and $B$ is Bernoulli on $\{-1,1\}$. This proves a conjecture of Forrester and Ipsen [13]. The results are obtained by the asymptotic analysis of a certain Meijer G-function.
</p>projecteuclid.org/euclid.ecp/1502416902_20171115221556Wed, 15 Nov 2017 22:15 ESTSignature inversion for monotone pathshttps://projecteuclid.org/euclid.ecp/1502762748<strong>Jiawei Chang</strong>, <strong>Nick Duffield</strong>, <strong>Hao Ni</strong>, <strong>Weijun Xu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 11 pp..</p><p><strong>Abstract:</strong><br/>
The aim of this article is to provide a simple sampling procedure to reconstruct any monotone path from its signature. For every $N$, we sample a lattice path of $N$ steps with weights given by the coefficient of the corresponding word in the signature. We show that these weights on lattice paths satisfy the large deviations principle. In particular, this implies that the probability of picking up a “wrong” path is exponentially small in $N$. The argument relies on a probabilistic interpretation of the signature for monotone paths.
</p>projecteuclid.org/euclid.ecp/1502762748_20171115221556Wed, 15 Nov 2017 22:15 ESTEnergy optimization for distributions on the sphere and improvement to the Welch boundshttps://projecteuclid.org/euclid.ecp/1502762749<strong>Yan Shuo Tan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
For any Borel probability measure on $\mathbb{R} ^n$, we may define a family of eccentricity tensors. This new notion, together with a tensorization trick, allows us to prove an energy minimization property for rotationally invariant probability measures. We use this theory to give a new proof of the Welch bounds, and to improve upon them for collections of real vectors. In addition, we are able to give elementary proofs for two theorems characterizing probability measures optimizing one-parameter families of energy integrals on the sphere. We are also able to explain why a phase transition occurs for optimizers of these two families.
</p>projecteuclid.org/euclid.ecp/1502762749_20171115221556Wed, 15 Nov 2017 22:15 ESTNonlinear filtering with degenerate noisehttps://projecteuclid.org/euclid.ecp/1502762750<strong>David Jaures Fotsa–Mbogne</strong>, <strong>Etienne Pardoux</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 14 pp..</p><p><strong>Abstract:</strong><br/>
This paper studies a new type of filtering problem, where the diffusion coefficient of the observation noise is strictly positive only in the interior of the bounded interval where observation takes its values. We derive a Zakai and a Kushner–Stratonovich equation, and prove uniqueness of the measure–valued solution of the Zakai equation.
</p>projecteuclid.org/euclid.ecp/1502762750_20171115221556Wed, 15 Nov 2017 22:15 ESTCutoff for Ramanujan graphs via degree inflationhttps://projecteuclid.org/euclid.ecp/1502762751<strong>Jonathan Hermon</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 10 pp..</p><p><strong>Abstract:</strong><br/>
Recently Lubetzky and Peres showed that simple random walks on a sequence of $d$-regular Ramanujan graphs $G_n=(V_n,E_n)$ of increasing sizes exhibit cutoff in total variation around the diameter lower bound $\frac{d} {d-2}\log _{d-1}|V_n| $. We provide a different argument under the assumption that for some $r(n) \gg 1$ the maximal number of simple cycles in a ball of radius $r(n)$ in $G_n$ is uniformly bounded in $n$.
</p>projecteuclid.org/euclid.ecp/1502762751_20171115221556Wed, 15 Nov 2017 22:15 EST