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 ESTA heat flow approach to the Godbillon-Vey class
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTApplication of stochastic flows to the sticky Brownian motion equation
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTRecurrence and transience properties of multi-dimensional diffusion processes in selfsimilar and semi-selfsimilar random environments
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTRecurrence of multiply-ended planar triangulations
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTUniversal large deviations for Kac polynomials
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTOn recurrence and transience of multivariate near-critical stochastic processes
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTYet another condition for absence of collisions for competing Brownian particles
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTStable limit theorem for $U$-statistic processes indexed by a random walk
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTNecessary and sufficient conditions for the $r$-excessive local martingales to be martingales
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTProduct space for two processes with independent increments under nonlinear expectations
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTThe set of connective constants of Cayley graphs contains a Cantor space
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTIndicable groups and $p_c<1$
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTFirst passage percolation on a hyperbolic graph admits bi-infinite geodesics
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTErratum: Optimal linear drift for the speed of convergence of an hypoelliptic diffusion
http://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_20170214040037Tue, 14 Feb 2017 04:00 ESTSelf-averaging sequences which fail to convergehttp://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_20170217220207Fri, 17 Feb 2017 22:02 ESTSome connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisetshttp://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_20170217220207Fri, 17 Feb 2017 22:02 ESTConvergence of complex martingales in the branching random walk: the boundaryhttp://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_20170228220131Tue, 28 Feb 2017 22:01 ESTRemarks on spectral gaps on the Riemannian path spacehttp://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_20170313220510Mon, 13 Mar 2017 22:05 EDTGalton-Watson probability contractionhttp://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_20170316040044Thu, 16 Mar 2017 04:00 EDTThe existence phase transition for two Poisson random fractal modelshttp://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_20170329220232Wed, 29 Mar 2017 22:02 EDTImproved bounds for the mixing time of the random-to-random shufflehttp://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_20170403220321Mon, 03 Apr 2017 22:03 EDTNote on A. Barbour’s paper on Stein’s method for diffusion approximationshttp://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_20170414220040Fri, 14 Apr 2017 22:00 EDTStein type characterization for $G$-normal distributionshttp://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_20170419040218Wed, 19 Apr 2017 04:02 EDTTwo observations on the capacity of the range of simple random walks on $\mathbb{Z} ^3$ and $\mathbb{Z} ^4$http://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_20170502220616Tue, 02 May 2017 22:06 EDTKesten’s incipient infinite cluster and quasi-multiplicativity of crossing probabilitieshttp://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_20170505220130Fri, 05 May 2017 22:01 EDTAn elementary approach to Gaussian multiplicative chaoshttp://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_20170511220044Thu, 11 May 2017 22:00 EDTAbout the constants in the Fuk-Nagaev inequalitieshttp://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_20170516040245Tue, 16 May 2017 04:02 EDTSurvival asymptotics for branching random walks in IID environmentshttp://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_20170524220232Wed, 24 May 2017 22:02 EDTThe frog model with drift on $\mathbb{R} $http://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_20170525220355Thu, 25 May 2017 22:03 EDTPiecewise constant local martingales with bounded numbers of jumpshttp://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_20170531220238Wed, 31 May 2017 22:02 EDTMarčenko-Pastur law for Kendall’s tauhttp://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_20170602220058Fri, 02 Jun 2017 22:00 EDTUniform convergence to the $Q$-processhttp://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_20170612220706Mon, 12 Jun 2017 22:07 EDTA note on Malliavin smoothness on the Lévy spacehttp://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_20170620220414Tue, 20 Jun 2017 22:04 EDTOn the relaxation rate of short chains of rotors interacting with Langevin thermostatshttp://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_20170620220414Tue, 20 Jun 2017 22:04 EDTA note on continuous-time stochastic approximation in infinite dimensionshttp://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_20170623040052Fri, 23 Jun 2017 04:00 EDTLarge deviations for biorthogonal ensembles and variational formulation for the Dykema-Haagerup distributionhttp://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_20170703040036Mon, 03 Jul 2017 04:00 EDTOn the Semi-classical Brownian Bridge Measurehttp://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_20170804040046Fri, 04 Aug 2017 04:00 EDTA functional limit theorem for excited random walkshttp://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_20170808220329Tue, 08 Aug 2017 22:03 EDTOn stochastic heat equation with measure initial datahttp://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_20170810220205Thu, 10 Aug 2017 22:02 EDTOn the real spectrum of a product of Gaussian matriceshttp://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_20170810220205Thu, 10 Aug 2017 22:02 EDTSignature inversion for monotone pathshttp://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_20170814220609Mon, 14 Aug 2017 22:06 EDTEnergy optimization for distributions on the sphere and improvement to the Welch boundshttp://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_20170814220609Mon, 14 Aug 2017 22:06 EDTNonlinear filtering with degenerate noisehttp://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_20170814220609Mon, 14 Aug 2017 22:06 EDTCutoff for Ramanujan graphs via degree inflationhttp://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_20170814220609Mon, 14 Aug 2017 22:06 EDT