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 ESTSmall deviations in lognormal Mandelbrot cascadeshttps://projecteuclid.org/euclid.ecp/1580202245<strong>Miika Nikula</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study small deviations in Mandelbrot cascades and some related models. Denoting by $Y$ the total mass variable of a Mandelbrot cascade generated by $W$, we show that if \[ \lim _{x \to 0} \frac{\log \log 1/\mathbb {P} (W \leq x)} {\log \log 1/x} = \gamma > 1, \] then the Laplace transform of $Y$ satisfies \[ \lim _{t \to \infty } \frac{\log \log 1/\mathbb {E}e^{-t Y}} {\log \log t} = \gamma . \] This implies the same estimate for $\mathbb{P} (Y \leq x)$ for small $x > 0$. As an application of the method, we prove a similar result for a variable arising as a total mass of a $\star $-scale invariant Gaussian multiplicative chaos measure.
</p>projecteuclid.org/euclid.ecp/1580202245_20200128040414Tue, 28 Jan 2020 04:04 ESTGeneralized scale functions of standard processes with no positive jumpshttps://projecteuclid.org/euclid.ecp/1580266866<strong>Kei Noba</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
As a generalization of scale functions of spectrally negative Lévy processes, we define generalized scale functions of general standard processes with no positive jumps. For this purpose, we utilize the excursion theory. Using the generalized scale functions, we study Laplace transforms of hitting times, potential measures and duality.
</p>projecteuclid.org/euclid.ecp/1580266866_20200128220128Tue, 28 Jan 2020 22:01 ESTGradient estimates and maximal dissipativity for the Kolmogorov operator in $\Phi ^{4}_{2}$https://projecteuclid.org/euclid.ecp/1580353227<strong>Giuseppe Da Prato</strong>, <strong>Arnaud Debussche</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 16 pp..</p><p><strong>Abstract:</strong><br/>
We consider the transition semigroup $P_{t}$ of the $\Phi ^{4}_{2}$ stochastic quantisation on the torus $\mathbb{T} ^{2}$ and prove the following new estimate (Theorem 3.10) \[ |DP_{t} \varphi (x)\cdot h|\le c\,t^{-\beta }\,|h|_{C^{-s}}\|\varphi \|_{0}\,(1+|x|_{C^{- \alpha }})^{\gamma }, \] for some $ \alpha ,\beta ,\gamma ,s$ positive. Thanks to this estimate, we show that cylindrical functions are a core for the corresponding Kolmogorov equation. Some consequences of this fact are discussed in a final remark.
</p>projecteuclid.org/euclid.ecp/1580353227_20200129220043Wed, 29 Jan 2020 22:00 ESTA remark on the smallest singular value of powers of Gaussian matriceshttps://projecteuclid.org/euclid.ecp/1580353228<strong>Han Huang</strong>, <strong>Konstantin Tikhomirov</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 8 pp..</p><p><strong>Abstract:</strong><br/>
Let $n,k\geq 1$ and let $G$ be the $n\times n$ random matrix with i.i.d. standard real Gaussian entries. We show that there are constants $c_{k},C_{k}>0$ depending only on $k$ such that the smallest singular value of $G^{k}$ satisfies \[ c_{k}\,t\leq{\mathbb {P}} \big \{s_{\min }(G^{k})\leq t^{k}\,n^{-1/2}\big \}\leq C_{k}\,t,\quad t\in (0,1], \] and, furthermore, \[ c_{k}/t\leq{\mathbb {P}} \big \{\|G^{-k}\|_{HS}\geq t^{k}\,n^{1/2}\big \}\leq C_{k}/t,\quad t\in [1,\infty ), \] where $\|\cdot \|_{HS}$ denotes the Hilbert–Schmidt norm.
</p>projecteuclid.org/euclid.ecp/1580353228_20200129220043Wed, 29 Jan 2020 22:00 ESTA note on costs minimization with stochastic target constraintshttps://projecteuclid.org/euclid.ecp/1580353229<strong>Yan Dolinsky</strong>, <strong>Benjamin Gottesman</strong>, <strong>Gurel-Gurevich Ori</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the minimization of the expected costs under stochastic constraint at the terminal time. The first and the main result says that for a power type of costs, the value function is the minimal positive solution of a second order semi-linear ordinary differential equation (ODE). Moreover, we establish the optimal control. In the second example we show that the case of exponential costs leads to a trivial optimal control.
</p>projecteuclid.org/euclid.ecp/1580353229_20200129220043Wed, 29 Jan 2020 22:00 ESTOccupation densities of ensembles of branching random walkshttps://projecteuclid.org/euclid.ecp/1580461331<strong>Steven P. Lalley</strong>, <strong>Si Tang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor $ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb{D} ([0, +\infty ), \mathcal{C} _{0}(\mathbb{R} ))$, as $N\to \infty $. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-valued stable-1/2 subordinator.
</p>projecteuclid.org/euclid.ecp/1580461331_20200131040215Fri, 31 Jan 2020 04:02 ESTDistributional analysis of the extra-clustering model with uniformly generated phylogenetic treeshttps://projecteuclid.org/euclid.ecp/1581130925<strong>Michael Fuchs</strong>, <strong>Chih-Hong Lee</strong>, <strong>Ariel R. Paningbatan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
The extra-clustering model for the group formation process of social animals was introduced by Durand, Blum and François. The model uses the relatedness of the animals, which is described by phylogenetic trees. If these trees are drawn from the Yule-Harding model, it was analyzed in recent work. Here, we analyze it for the uniform model, which is the other widely-studied model on phylogenetics trees. More precisely, we derive moments and limit laws for the number of groups, the number of groups of fixed size and the largest group size. Our results show that, independent of the probability of extra-clustering, there is on average only a finite number of groups, one of which is very large whereas all others are small. This behavior considerably differs from the Yule-Harding case, where the finiteness of the number of groups is dependent on the extra-clustering probability.
</p>projecteuclid.org/euclid.ecp/1581130925_20200207220231Fri, 07 Feb 2020 22:02 ESTBounds on the probability of radically different opinionshttps://projecteuclid.org/euclid.ecp/1581325211<strong>Krzysztof Burdzy</strong>, <strong>Jim Pitman</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
We establish bounds on the probability that two different agents, who share an initial opinion expressed as a probability distribution on an abstract probability space, given two different sources of information, may come to radically different opinions regarding the conditional probability of the same event.
</p>projecteuclid.org/euclid.ecp/1581325211_20200210040024Mon, 10 Feb 2020 04:00 ESTA quantitative McDiarmid’s inequality for geometrically ergodic Markov chainshttps://projecteuclid.org/euclid.ecp/1581325212<strong>Antoine Havet</strong>, <strong>Matthieu Lerasle</strong>, <strong>Eric Moulines</strong>, <strong>Elodie Vernet</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 11 pp..</p><p><strong>Abstract:</strong><br/>
We state and prove a quantitative version of the bounded difference inequality for geometrically ergodic Markov chains. Our proof uses the same martingale decomposition as [2] but, compared to this paper, the exact coupling argument is modified to fill a gap between the strongly aperiodic case and the general aperiodic case.
</p>projecteuclid.org/euclid.ecp/1581325212_20200210040024Mon, 10 Feb 2020 04:00 ESTLarge deviations related to the law of the iterated logarithm for Itô diffusionshttps://projecteuclid.org/euclid.ecp/1581995086<strong>Stefan Gerhold</strong>, <strong>Christoph Gerstenecker</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 11 pp..</p><p><strong>Abstract:</strong><br/>
When a Brownian motion is scaled according to the law of the iterated logarithm, its supremum converges to one as time tends to zero. Upper large deviations of the supremum process can be quantified by writing the problem in terms of hitting times and applying a result of Strassen (1967) on hitting time densities. We extend this to a small-time large deviations principle for the supremum of scaled Itô diffusions, using as our main tool a refinement of Strassen’s result due to Lerche (1986).
</p>projecteuclid.org/euclid.ecp/1581995086_20200217220455Mon, 17 Feb 2020 22:04 ESTPivotality versus noise stability for monotone transitive functionshttps://projecteuclid.org/euclid.ecp/1582167853<strong>Pál Galicza</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 6 pp..</p><p><strong>Abstract:</strong><br/>
We construct a noise stable sequence of transitive, monotone increasing Boolean functions $f_{n}: \{-1,1\}^{k_{n}} \longrightarrow \{-1,1\}$ which admit many pivotals with high probability. We show that such a sequence is volatile as well, and thus it is also an example of a volatile and noise stable sequence of transitive, monotone functions.
</p>projecteuclid.org/euclid.ecp/1582167853_20200219220426Wed, 19 Feb 2020 22:04 ESTWeak monotone rearrangement on the linehttps://projecteuclid.org/euclid.ecp/1582254306<strong>Julio Backhoff-Veraguas</strong>, <strong>Mathias Beiglböck</strong>, <strong>Gudmund Pammer</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 16 pp..</p><p><strong>Abstract:</strong><br/>
Weak optimal transport has been recently introduced by Gozlan et al. The original motivation stems from the theory of geometric inequalities; further applications concern numerics of martingale optimal transport and stability in mathematical finance.
In this note we provide a complete geometric characterization of the weak version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain.
</p>projecteuclid.org/euclid.ecp/1582254306_20200220220524Thu, 20 Feb 2020 22:05 ESTA rotor configuration with maximum escape ratehttps://projecteuclid.org/euclid.ecp/1582686118<strong>Swee Hong Chan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 5 pp..</p><p><strong>Abstract:</strong><br/>
The rotor walk is a deterministic analogue of the simple random walk. For any given graph, we construct a rotor configuration for which the escape rate of the corresponding rotor walk is equal to the escape rate of the simple random walk, and thus answer a question of Florescu, Ganguly, Levine, and Peres (2014).
</p>projecteuclid.org/euclid.ecp/1582686118_20200225220205Tue, 25 Feb 2020 22:02 ESTRemarks on Gross’ technique for obtaining a conformal Skorohod embedding of planar Brownian motionhttps://projecteuclid.org/euclid.ecp/1582945213<strong>Maher Boudabra</strong>, <strong>Greg Markowsky</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
In [7] it was proved that, given a distribution $\mu $ with zero mean and finite second moment, there exists a simply connected domain $\Omega $ such that if $Z_{t}$ is a standard planar Brownian motion, then $\mathcal{R} e(Z_{\tau _{\Omega }})$ has the distribution $\mu $, where $\tau _{\Omega }$ denotes the exit time of $Z_{t}$ from $\Omega $. In this note, we extend this method to prove that if $\mu $ has a finite $p$-th moment then the first exit time $\tau _{\Omega }$ from $\Omega $ has a finite moment of order $\frac{p} {2}$. We also prove a uniqueness principle for this construction, and use it to give several examples.
</p>projecteuclid.org/euclid.ecp/1582945213_20200228220023Fri, 28 Feb 2020 22:00 ESTA ratio inequality for nonnegative martingales and their differential subordinateshttps://projecteuclid.org/euclid.ecp/1582945214<strong>Adam Osękowski</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
We prove a sharp inequality for the ratio of the maximal functions of nonnegative martingales and their differential subordinates. An application in the theory of weighted inequalities is given.
</p>projecteuclid.org/euclid.ecp/1582945214_20200228220023Fri, 28 Feb 2020 22:00 ESTCoexistence in chase-escapehttps://projecteuclid.org/euclid.ecp/1583290992<strong>Rick Durrett</strong>, <strong>Matthew Junge</strong>, <strong>Si Tang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 14 pp..</p><p><strong>Abstract:</strong><br/>
We study a competitive stochastic growth model called chase-escape in which red particles spread to adjacent uncolored sites and blue particles only to adjacent red sites. Red particles are killed when blue occupies the same site. If blue has rate-1 passage times and red rate-$\lambda $, a phase transition occurs for the probability red escapes to infinity on $\mathbb{Z} ^{d}$, $d$-ary trees, and the ladder graph $\mathbb{Z} \times \{0,1\}$. The result on the tree was known, but we provide a new, simpler calculation of the critical value, and observe that it is a lower bound for a variety of graphs. We conclude by showing that red can be stochastically slower than blue, but still escape with positive probability for large enough $d$ on oriented $\mathbb{Z} ^{d}$ with passage times that resemble Bernoulli bond percolation.
</p>projecteuclid.org/euclid.ecp/1583290992_20200303220321Tue, 03 Mar 2020 22:03 EST$V$-geometrical ergodicity of Markov kernels via finite-rank approximationshttps://projecteuclid.org/euclid.ecp/1584151270<strong>Loïc Hervé</strong>, <strong>James Ledoux</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the $V$-geometrical ergodicity of a general Markov kernel $P$, which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of $P$ by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in $V$-geometrical ergodicity.
</p>projecteuclid.org/euclid.ecp/1584151270_20200313220117Fri, 13 Mar 2020 22:01 EDTThe contact process on periodic treeshttps://projecteuclid.org/euclid.ecp/1585101897<strong>Xiangying Huang</strong>, <strong>Rick Durrett</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
A little over 25 years ago Pemantle [6] pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values $\lambda _{1}$ and $\lambda _{2}$ for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generations is $(n,a_{1},\ldots , a_{k})$ with $\max _{i} a_{i} \le Cn^{1-\delta }$ and $\log (a_{1} \cdots a_{k})/\log n \to b$ as $n\to \infty $. We show that the critical value for local survival is asymptotically $\sqrt{c (\log n)/n} $ where $c=(k-b)/2$. This supports Pemantle’s claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.
</p>projecteuclid.org/euclid.ecp/1585101897_20200324220518Tue, 24 Mar 2020 22:05 EDTPhase transitions for chase-escape models on Poisson–Gilbert graphshttps://projecteuclid.org/euclid.ecp/1585188174<strong>Alexander Hinsen</strong>, <strong>Benedikt Jahnel</strong>, <strong>Elie Cali</strong>, <strong>Jean-Philippe Wary</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 14 pp..</p><p><strong>Abstract:</strong><br/>
We present results on phase transitions of local and global survival in a two-species model on Poisson–Gilbert graphs. Initially, there is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system. The graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.
</p>projecteuclid.org/euclid.ecp/1585188174_20200325220300Wed, 25 Mar 2020 22:03 EDTCoexistence in a random growth model with competitionhttps://projecteuclid.org/euclid.ecp/1585274664<strong>Shane Turnbull</strong>, <strong>Amanda Turner</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 14 pp..</p><p><strong>Abstract:</strong><br/>
We consider a variation of the Hastings-Levitov model HL(0) for random growth in which the growing cluster consists of two competing regions. We allow the size of successive particles to depend both on the region in which the particle is attached, and the harmonic measure carried by that region. We identify conditions under which one can ensure coexistence of both regions. In particular, we consider whether it is possible for the process giving the relative harmonic measures of the regions to converge to a non-trivial ergodic limit.
</p>projecteuclid.org/euclid.ecp/1585274664_20200326220430Thu, 26 Mar 2020 22:04 EDTDonsker’s theorem in Wasserstein-1 distancehttps://projecteuclid.org/euclid.ecp/1585620027<strong>Laure Coutin</strong>, <strong>Laurent Decreusefond</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
We compute the Wassertein-1 (or Kantorovitch-Rubinstein) distance between a random walk in $\mathbf{R} ^{d}$ and the Brownian motion. The proof is based on a new estimate of the modulus of continuity of the solution of the Stein’s equation. As an application, we can evaluate the rate of convergence towards the local time at 0 of the Brownian motion and to a Brownian bridge.
</p>projecteuclid.org/euclid.ecp/1585620027_20200330220036Mon, 30 Mar 2020 22:00 EDTOn the central limit theorem for the two-sided descent statistics in Coxeter groupshttps://projecteuclid.org/euclid.ecp/1585620028<strong>Valentin Féray</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 6 pp..</p><p><strong>Abstract:</strong><br/>
In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups $W_{n}$ for which the two-sided descent statistics on a uniform random element of $W_{n}$ is asymptotically normal. Recently, Brück and Röttger provided an almost-complete answer, assuming some regularity condition on the sequence $W_{n}$. In this note, we provide a shorter proof of their result, which does not require any regularity condition. The main new proof ingredient is the use of the second Wasserstein distance on probability distributions, based on the work of Mallows (Ann. Math. Statist., 1972).
</p>projecteuclid.org/euclid.ecp/1585620028_20200330220036Mon, 30 Mar 2020 22:00 EDTExtensions of Brownian motion to a family of Grushin-type singularitieshttps://projecteuclid.org/euclid.ecp/1586332822<strong>Ugo Boscain</strong>, <strong>Robert W. Neel</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. This gives a quick proof and clear intuition for the fact that heat can only cross the singularity for an intermediate range of the parameter. When crossing is possible and the singularity consists of one point, we give a complete description of these diffusions, and we describe a “best” extension, which respects the isometry group of the surface and also realizes the unique symmetric one-point extension of the Brownian motion, in the sense of Chen-Fukushima. This extension, however, does not correspond to the bridging extension, which was introduced by Boscain-Prandi, when they previously considered self-adjoint extensions of the Laplace-Beltrami operator on the Riemannian part for these surfaces. We clarify that several of the extensions they considered induce diffusions that are carried by the Martin compactification at the singularity, which is much larger than the (one-point) metric completion. In the case when the singularity is more than one-point, a complete classification of diffusions extending Brownian motion would be unwieldy. Nonetheless, we again describe a “best” extension which respects the isometry group, and in this case, this diffusion corresponds to the bridging extension. A prominent role is played by Bessel processes (of every real dimension) and the classical theory of one-dimensional diffusions and their boundary conditions.
</p>projecteuclid.org/euclid.ecp/1586332822_20200408040029Wed, 08 Apr 2020 04:00 EDTProbabilistic interpretation of HJB equations by the representation theorem for generators of BSDEshttps://projecteuclid.org/euclid.ecp/1586419347<strong>Lishun Xiao</strong>, <strong>Shengjun Fan</strong>, <strong>Dejian Tian</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 10 pp..</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of backward stochastic differential equations. The key idea of our approach for proving this interpretation lies in the identity between solutions and generators given by the representation theorem. Compared with existing methods, our approach seems to be a feasible unified method for different frameworks and be more applicable to general settings. This can also be regarded as a new application of such representation theorem.
</p>projecteuclid.org/euclid.ecp/1586419347_20200409040233Thu, 09 Apr 2020 04:02 EDTErratum: Practical criteria for $R$-positive recurrence of unbounded semigroupshttps://projecteuclid.org/euclid.ecp/1586916022<strong>Nicolas Champagnat</strong>, <strong>Denis Villemonais</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 2 pp..</p><p><strong>Abstract:</strong><br/>
The proof of the second convergence of Theorem 2.1 provided in the original paper relies on properties of the so called Q-process, which is not, in general, defined on the whole state space. Hence the inequality is only proved for some restricted sets of functions and measures. This weakness can be easily repaired and we provide here a proof of the inequality that applies in full generality.
</p>projecteuclid.org/euclid.ecp/1586916022_20200414220030Tue, 14 Apr 2020 22:00 EDTOn Absence of disorder chaos for spin glasses on $\mathbb{Z} ^{d}$https://projecteuclid.org/euclid.ecp/1587693906<strong>Louis-Pierre Arguin</strong>, <strong>Jack Hanson</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
We identify simple mechanisms that prevent a strong form of disorder chaos for edge overlap of the Ising spin glass model on $\mathbb{Z} ^{d}$. This was first shown by Chatterjee in the case of Gaussian couplings. We present three proofs of the theorem for general couplings with continuous distribution based on the presence in the coupling realization of stabilizing features of positive density.
</p>projecteuclid.org/euclid.ecp/1587693906_20200423220528Thu, 23 Apr 2020 22:05 EDTStability for Hawkes processes with inhibitionhttps://projecteuclid.org/euclid.ecp/1588039621<strong>Mads Bonde Raad</strong>, <strong>Eva Löcherbach</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 9 pp..</p><p><strong>Abstract:</strong><br/>
We consider a multivariate non-linear Hawkes process in a multi-class setup where particles are organised within two populations of possibly different sizes, such that one of the populations acts excitatory on the system while the other population acts inhibitory on the system. The goal of this note is to present a class of Hawkes Processes with stable dynamics without assumptions on the spectral radius of the associated weight function matrix. This illustrates how inhibition in a Hawkes system significantly affects the stability properties of the system.
</p>projecteuclid.org/euclid.ecp/1588039621_20200427220710Mon, 27 Apr 2020 22:07 EDTAnchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuoushttps://projecteuclid.org/euclid.ecp/1588125634<strong>Barbara Dembin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
We consider the standard model of i.i.d. first passage percolation on $\mathbb{Z} ^{d}$ given a distribution $G$ on $\mathbb{R} _{+}$. We consider a cube oriented in the direction $\overrightarrow{v} $ whose sides have length $n$. We study the maximal flow from the top half to the bottom half of the boundary of this cube. We already know that the maximal flow renormalized by $n^{d-1}$ converges towards the flow constant $\nu _{G}(\overrightarrow{v} )$. We prove here that the map $p\mapsto \nu _{p\delta _{1}+(1-p)\delta _{0}}$ is Lipschitz continuous on all intervals $[p_{0},p_{1}]\subset (p_{c}(d),1)$ where $p_{c}(d)$ denotes the critical parameter for i.i.d. bond percolation on $\mathbb{Z} ^{d}$. For $p>p_{c}(d)$, we know that there exists almost surely a unique infinite open cluster $\mathcal{C} _{p}$ [8]. We are interested in the regularity properties in $p$ of the anchored isoperimetric profile of the infinite cluster $\mathcal{C} _{p}$. For $d\geq 2$, using the result on the regularity of the flow constant, we prove here that the anchored isoperimetric profile defined in [4] is Lipschitz continuous on all intervals $[p_{0},p_{1}]\subset (p_{c}(d),1)$.
</p>projecteuclid.org/euclid.ecp/1588125634_20200428220044Tue, 28 Apr 2020 22:00 EDTPolynomial rate of convergence to the Yaglom limit for Brownian motion with drifthttps://projecteuclid.org/euclid.ecp/1588903421<strong>William Oçafrain</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
This paper deals with the rate of convergence in $1$-Wasserstein distance of the marginal law of a Brownian motion with drift conditioned not to have reached $0$ towards the Yaglom limit of the process. In particular it is shown that, for a wide class of initial measures including probability measures with compact support, the Wasserstein distance decays asymptotically as $1/t$. Likewise, this speed of convergence is recovered for the convergence of marginal laws conditioned not to be absorbed up to a horizon time towards the Bessel-$3$ process, when the horizon time tends to infinity.
</p>projecteuclid.org/euclid.ecp/1588903421_20200507220401Thu, 07 May 2020 22:04 EDTNew asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficientshttps://projecteuclid.org/euclid.ecp/1588903422<strong>Pautrel Thibault</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
We consider random trigonometric polynomials of the form \[ f_{n}(t):=\frac{1} {\sqrt{n} } \sum _{k=1}^{n}a_{k} \cos (k t)+b_{k} \sin (k t), \] where $(a_{k})_{k\geq 1}$ and $(b_{k})_{k\geq 1}$ are two independent stationary Gaussian processes with the same correlation function $\rho : k \mapsto \cos (k\alpha )$, with $\alpha \geq 0$. We show that the asymptotics of the expected number of real zeros differ from the universal one $\frac{2} {\sqrt{3} }$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $\varepsilon >0$, for all $\ell \in (\sqrt{2} ,2]$, there exists $\alpha \geq 0$ and $n\geq 1$ large enough such that \[ \left |\frac{\mathbb {E} \left [\mathcal {N}(f_{n},[0,2\pi ])\right ]} {n}-\ell \right |\leq \varepsilon , \] where $\mathcal{N} (f_{n},[0,2\pi ])$ denotes the number of real zeros of the function $f_{n}$ in the interval $[0,2\pi ]$. Therefore, this result provides the first example where the expected number of real zeros does not converge as $n$ goes to infinity by exhibiting a whole range of possible subsequential limits ranging from $\sqrt{2} $ to 2.
</p>projecteuclid.org/euclid.ecp/1588903422_20200507220401Thu, 07 May 2020 22:04 EDTOn the volume of the shrinking branching Brownian sausagehttps://projecteuclid.org/euclid.ecp/1589335619<strong>Mehmet Öz</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
The branching Brownian sausage in $\mathbb{R} ^{d}$ was defined in [4] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R} ^{d}$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated branching Brownian sausage (BBM-sausage) with radius exponentially shrinking in time. Using a previous result on the density of the support of BBM, and some well-known results on the classical Wiener sausage and Brownian hitting probabilities, we obtain almost sure limit theorems as time tends to infinity on the volume of the shrinking BBM-sausage in all dimensions.
</p>projecteuclid.org/euclid.ecp/1589335619_20200512220712Tue, 12 May 2020 22:07 EDTMetrics on sets of interval partitions with diversityhttps://projecteuclid.org/euclid.ecp/1591668057<strong>Noah Forman</strong>, <strong>Soumik Pal</strong>, <strong>Douglas Rizzolo</strong>, <strong>Matthias Winkel</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 16 pp..</p><p><strong>Abstract:</strong><br/>
We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals. Further restricting to interval partitions with $\alpha $-diversity, we then adjust the metric to incorporate diversities. We show that this second metric space is Lusin. An important feature of this topology is that path-continuity in this topology implies the continuous evolution of diversities. This is important in related work on tree-valued stochastic processes where diversities are branch lengths.
</p>projecteuclid.org/euclid.ecp/1591668057_20200608220111Mon, 08 Jun 2020 22:01 EDTA martingale approach for Pólya urn processeshttps://projecteuclid.org/euclid.ecp/1591840901<strong>Lucile Laulin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
This paper is devoted to a direct martingale approach for Pólya urn models asymptotic behaviour. A Pólya process is said to be small when the ratio of its replacement matrix eigenvalues is less than or equal to $1/2$, otherwise it is called large. We find again some well-known results on the asymptotic behaviour for small and large urn processes. We also provide new almost sure properties for small urn processes.
</p>projecteuclid.org/euclid.ecp/1591840901_20200610220200Wed, 10 Jun 2020 22:02 EDTOn the CLT for additive functionals of Markov chainshttps://projecteuclid.org/euclid.ecp/1592272818<strong>Magda Peligrad</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 10 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we study the additive functionals of Markov chains via conditioning with respect to both past and future of the chain. We shall point out new sufficient projective conditions, which assure that the variance of partial sums of $n$ consecutive random variables of a stationary Markov chain is linear in $n$. The paper also addresses the central limit theorem problem and is listing several open questions.
</p>projecteuclid.org/euclid.ecp/1592272818_20200615220028Mon, 15 Jun 2020 22:00 EDTLower large deviations for geometric functionalshttps://projecteuclid.org/euclid.ecp/1592272819<strong>Christian Hirsch</strong>, <strong>Benedikt Jahnel</strong>, <strong>András Tóbiás</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson–Voronoi cells, as well as power-weighted edge lengths in the random geometric, $k$-nearest neighbor and relative neighborhood graph.
</p>projecteuclid.org/euclid.ecp/1592272819_20200615220028Mon, 15 Jun 2020 22:00 EDTConformal Skorokhod embeddings and related extremal problemshttps://projecteuclid.org/euclid.ecp/1592446014<strong>Phanuel Mariano</strong>, <strong>Hugo Panzo</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 11 pp..</p><p><strong>Abstract:</strong><br/>
The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset \mathbb {C}$ whose exit time embeds a given probability distribution $\mu $ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding $\mu $ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb {U}$ that attains the lowest possible principal Dirichlet eigenvalue over all domains solving the CSEP for the uniform distribution on $[-1,1]$. Remarkably, the boundary of $\mathbb {U}$ is related to the Grim Reaper translating solution to the curve shortening flow in the plane. The novel tool used in the proof of the sharp lower bound is a precise relationship between the widths of the orthogonal projections of a simply connected planar domain and the support of its harmonic measure that we develop in the paper. The upper bound relies on spectral bounds for the torsion function which have recently appeared in the literature.
</p>projecteuclid.org/euclid.ecp/1592446014_20200617220704Wed, 17 Jun 2020 22:07 EDTAnalyticity for rapidly determined properties of Poisson Galton–Watson treeshttps://projecteuclid.org/euclid.ecp/1592553612<strong>Yuval Peres</strong>, <strong>Andrew Swan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 8 pp..</p><p><strong>Abstract:</strong><br/>
Let $T_{\lambda }$ be a Galton–Watson tree with Poisson($\lambda $) offspring, and let $A$ be a tree property. In this paper, we are concerned with the regularity of the function $\mathbb {P}_{\lambda }(A)\coloneqq \mathbb {P}(T_{\lambda }\models A)$. We show that if a property $A$ can be uniformly approximated by a sequence of properties $\{A_{k}\}$, depending only on the first $k$ vertices in the breadth first exploration of the tree, with a bound in probability of $\mathbb {P}_{\lambda }(A\triangle A_{k}) \le Ce^{-ck}$ over an interval $I = (\lambda _{0}, \lambda _{1})$, then $\mathbb {P}_{\lambda }(A)$ is real analytic in $\lambda $ for $\lambda \in I$. We also present some applications of our results, particularly to properties that are not expressible in first order logic on trees.
</p>projecteuclid.org/euclid.ecp/1592553612_20200619040026Fri, 19 Jun 2020 04:00 EDTKilled rough super-Brownian motionhttps://projecteuclid.org/euclid.ecp/1592877699<strong>Tommaso Cornelis Rosati</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
This article concerns the construction of a continuous branching process in a random, time-independent environment, on finite volume. The backbone of this study is the convergence of discrete approximations of the parabolic Anderson model (PAM) on a box with Dirichlet boundary conditions. This is a companion paper to [9].
</p>projecteuclid.org/euclid.ecp/1592877699_20200622220149Mon, 22 Jun 2020 22:01 EDTA coupling proof of convex ordering for compound distributionshttps://projecteuclid.org/euclid.ecp/1593569165<strong>Jean Bérard</strong>, <strong>Nicolas Juillet</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 9 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we give an alternative proof of the fact that, when compounding a nonnegative probability distribution, convex ordering between the distributions of the number of summands implies convex ordering between the resulting compound distributions. Although this is a classical textbook result in risk theory, our proof exhibits a concrete coupling between the compound distributions being compared, using the representation of one-period discrete martingale laws as a mixture of the corresponding extremal measures.
</p>projecteuclid.org/euclid.ecp/1593569165_20200630220616Tue, 30 Jun 2020 22:06 EDTIntegration by parts formulae for the laws of Bessel bridges via hypergeometric functionshttps://projecteuclid.org/euclid.ecp/1593569166<strong>Henri Elad Altman</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 11 pp..</p><p><strong>Abstract:</strong><br/>
In this article, we extend the integration by parts formulae (IbPF) for the laws of Bessel bridges recently obtained in [2] to linear functionals. Our proof relies on properties of hypergeometric functions, thus providing a new interpretation of these formulae.
</p>projecteuclid.org/euclid.ecp/1593569166_20200630220616Tue, 30 Jun 2020 22:06 EDTA lower bound for point-to-point connection probabilities in critical percolationhttps://projecteuclid.org/euclid.ecp/1593569167<strong>J. van den Berg</strong>, <strong>H. Don</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 9 pp..</p><p><strong>Abstract:</strong><br/>
Consider critical site percolation on $\mathbb{Z} ^{d}$ with $d \geq 2$. We prove a lower bound of order $n^{- d^{2}}$ for point-to-point connection probabilities, where $n$ is the distance between the points.
Most of the work in our proof concerns a ‘construction’ which finally reduces the problem to a topological one. This is then solved by applying a topological fact, Lemma 2.12 below, which follows from Brouwer’s fixed point theorem.
Our bound improves the lower bound with exponent $2 d (d-1)$, used by Cerf in 2015 [1] to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in itself, our result gives a small improvement of the bound on the two-arm exponent found by Cerf.
</p>projecteuclid.org/euclid.ecp/1593569167_20200630220616Tue, 30 Jun 2020 22:06 EDTIntermittency for the parabolic Anderson model of Skorohod type driven by a rough noisehttps://projecteuclid.org/euclid.ecp/1594713622<strong>Nicholas Ma</strong>, <strong>David Nualart</strong>, <strong>Panqiu Xia</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 10 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter $H\in (0,1/2)$. By using the Feynman-Kac representation for the $L^{p}(\Omega )$ moments of the solution, we find the upper and lower bounds for the moments.
</p>projecteuclid.org/euclid.ecp/1594713622_20200714040034Tue, 14 Jul 2020 04:00 EDTOn the strict value of the non-linear optimal stopping problemhttps://projecteuclid.org/euclid.ecp/1595037888<strong>Miryana Grigorova</strong>, <strong>Peter Imkeller</strong>, <strong>Youssef Ouknine</strong>, <strong>Marie-Claire Quenez</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 9 pp..</p><p><strong>Abstract:</strong><br/>
We address the non-linear strict value problem in the case of a general filtration and a completely irregular pay-off process $(\xi _{t})$. While the value process $(V_{t})$ of the non-linear problem is only right-uppersemicontinuous, we show that the strict value process $(V^{+}_{t})$ is necessarily right-continuous. Moreover, the strict value process $(V_{t}^{+})$ coincides with the process of right-limits $(V_{t+})$ of the value process. As an auxiliary result, we obtain that a strong non-linear $f$-supermartingale is right-continuous if and only if it is right-continuous along stopping times in conditional $f$-expectation.
</p>projecteuclid.org/euclid.ecp/1595037888_20200717220459Fri, 17 Jul 2020 22:04 EDTLoad balancing under $d$-thinninghttps://projecteuclid.org/euclid.ecp/1578020668<strong>Ohad Noy Feldheim</strong>, <strong>Jiange Li</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
In the classical balls-and-bins model, $m$ balls are allocated into $n$ bins one by one uniformly at random. In this note, we consider the $d$-thinning variant of this model, in which the process is regulated in an on-line fashion as follows. For each ball, after a random bin has been selected, an overseer may decide, based on all previous history, whether to accept this bin or not. However, one of every $d$ consecutive suggested bins must be accepted. The maximum load of this setting is the number of balls in the most loaded bin. We show that after $\Theta (n)$ balls have been allocated, the least maximum load achievable with high probability is $(d+o(1))\sqrt [d]{\frac{d\log n} {\log \log n}}$. This should be compared with the related $d$-choice setting, in which the optimal maximum load achievable with high probability is $\frac{\log \log n} {\log d}+O(1)$.
</p>projecteuclid.org/euclid.ecp/1578020668_20200722220425Wed, 22 Jul 2020 22:04 EDTThe initial set in the frog model is irrelevanthttps://projecteuclid.org/euclid.ecp/1595469840<strong>Maria Deijfen</strong>, <strong>Sebastian Rosengrenï</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 7 pp..</p><p><strong>Abstract:</strong><br/>
In this note, we consider the frog model on $\mathbb {Z}^{d}$ and a two-type version of it with two types of particles. For the one-type model, we show that the asymptotic shape does not depend on the initially activated set and the configuration there. For the two-type model, we show that the possibility for the types to coexist in that both of them activate infinitely many particles does not depend on the choice of the initially activated sets and the configurations there.
</p>projecteuclid.org/euclid.ecp/1595469840_20200722220425Wed, 22 Jul 2020 22:04 EDTVanishing of the anchored isoperimetric profile in bond percolation at $p_{c}$https://projecteuclid.org/euclid.ecp/1578106940<strong>Raphaël Cerf</strong>, <strong>Barbara Dembin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 7 pp..</p><p><strong>Abstract:</strong><br/>
We consider the anchored isoperimetric profile of the infinite open cluster, defined for $p>p_{c}$, whose existence has been recently proved in [3]. We extend adequately the definition for $p=p_{c}$, in finite boxes. We prove a partial result which implies that, if the limit defining the anchored isoperimetric profile at $p_{c}$ exists, it has to vanish.
</p>projecteuclid.org/euclid.ecp/1578106940_20200730040138Thu, 30 Jul 2020 04:01 EDTStrong Feller property and continuous dependence on initial data for one-dimensional stochastic differential equations with Hölder continuous coefficientshttps://projecteuclid.org/euclid.ecp/1578387617<strong>Hua Zhang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 10 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, under the assumption of Hölder continuous coefficients, we prove the strong Feller property and continuous dependence on initial data for the solution to one-dimensional stochastic differential equations whose proof are based on the technique of local time, coupling method and Girsanov’s transform.
</p>projecteuclid.org/euclid.ecp/1578387617_20200730040138Thu, 30 Jul 2020 04:01 EDTAsymptotics of Schur functions on almost staircase partitionshttps://projecteuclid.org/euclid.ecp/1596096082<strong>Zhongyang Li</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 13 pp..</p><p><strong>Abstract:</strong><br/>
We study the asymptotics of Schur polynomials with partitions $\lambda $ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),\ldots ,(m-1),0)$ by at most one component at the beginning as $N\rightarrow \infty $, for a positive integer $m\ge 1$ independent of $N$. By applying either determinant formulas or integral representations for Schur functions, we show that $\frac {1}{N}\log \frac {s_{\lambda }(u_{1},\ldots ,u_{k}, x_{k+1},\ldots ,x_{N})}{s_{\lambda }(x_{1},\ldots ,x_{N})}$ converges to a sum of $k$ single-variable holomorphic functions, each of which depends on the variable $u_{i}$ for $1\leq i\leq k$, when there are only finitely many distinct $x_{i}$’s and each $u_{i}$ is in a neighborhood of $x_{i}$, as $N\rightarrow \infty $. The results are related to the law of large numbers and central limit theorem for the dimer configurations on contracting square-hexagon lattices with certain boundary conditions.
</p>projecteuclid.org/euclid.ecp/1596096082_20200730040138Thu, 30 Jul 2020 04:01 EDTCharacterising random partitions by random colouringhttps://projecteuclid.org/euclid.ecp/1578906086<strong>Jakob E. Björnberg</strong>, <strong>Cécile Mailler</strong>, <strong>Peter Mörters</strong>, <strong>Daniel Ueltschi</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 12 pp..</p><p><strong>Abstract:</strong><br/>
Let $(X_{1},X_{2},...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_{i}\geq 0$ and $\sum _{i\geq 1} X_{i}=1$, and let $(\varepsilon _{1}, \varepsilon _{2},...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the random variable $Z =\sum _{i\geq 1} \varepsilon _{i} X_{i}$. The question addressed in this article is: Knowing the distribution of $Z$ for some fixed $p\in (0,1)$, what can we infer about the random partition $(X_{1}, X_{2},...)$? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter $p$ is not equal to $\nicefrac {1}{2}$.
</p>projecteuclid.org/euclid.ecp/1578906086_20200731220238Fri, 31 Jul 2020 22:02 EDTThe remainder in the renewal theoremhttps://projecteuclid.org/euclid.ecp/1580180423<strong>Ron Doney</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 8 pp..</p><p><strong>Abstract:</strong><br/>
If the step distribution in a renewal process has finite mean and regularly varying tail with index $-\alpha $, $1<\alpha <2$, the first two terms in the asymptotic expansion of the renewal function have been known for many years. Here we show that, without making any additional assumptions, it is possible to give, in all cases except for $\alpha =3/2$, the exact asymptotic behaviour of the next term. In the case $\alpha =3/2$ the result is exact to within a slowly varying correction. Similar results are shown to hold in the random walk case.
</p>projecteuclid.org/euclid.ecp/1580180423_20200731220238Fri, 31 Jul 2020 22:02 EDTPractical criteria for $R$-positive recurrence of unbounded semigroupshttps://projecteuclid.org/euclid.ecp/1580180424<strong>Nicolas Champagnat</strong>, <strong>Denis Villemonais</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 11 pp..</p><p><strong>Abstract:</strong><br/>
The goal of this note is to show how recent results on the theory of quasi-stationary distributions allow us to deduce general criteria for the geometric convergence of normalized unbounded semigroups.
</p>projecteuclid.org/euclid.ecp/1580180424_20200731220238Fri, 31 Jul 2020 22:02 EDTOn the long-time behaviour of McKean-Vlasov pathshttps://projecteuclid.org/euclid.ecp/1596247347<strong>K. Bashiri</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 25, 14 pp..</p><p><strong>Abstract:</strong><br/>
It is well-known that, in a certain parameter regime, the so-called McKean-Vlasov evolution $(\mu _{t})_{t\in [0,\infty )}$ admits exactly three stationary states . In this paper we study the long-time behaviour of the flow $(\mu _{t})_{t\in [0,\infty )}$ in this regime. The main result is that, for any initial measure $\mu _{0}$, the flow $(\mu _{t})_{t\in [0,\infty )}$ converges to a stationary state as $t\rightarrow \infty $ (see Theorem 1.2). Moreover, we show that if the energy of the initial measure is below some critical threshold, then the limiting stationary state can be identified (see Proposition 1.3). Finally, we also show some topological properties of the basins of attraction of the McKean-Vlasov evolution (see Proposition 1.4). The proofs are based on the representation of $(\mu _{t})_{t\in [0,\infty )}$ as a Wasserstein gradient flow .
Some results of this paper are not entirely new. The main contribution here is to show that the Wasserstein framework provides short and elegant proofs for these results. However, up to the author’s best knowledge, the statement on the topological properties of the basins of attraction (Proposition 1.4) is a new result.
</p>projecteuclid.org/euclid.ecp/1596247347_20200731220238Fri, 31 Jul 2020 22:02 EDT