Mostrando recursos 1 - 20 de 919

  1. Note on A. Barbour’s paper on Stein’s method for diffusion approximations

    Kasprzak, Mikołaj J.; Duncan, Andrew B.; Vollmer, Sebastian J.
    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...

  2. Improved bounds for the mixing time of the random-to-random shuffle

    Qin, Chuan; Morris, Ben
    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.

  3. The existence phase transition for two Poisson random fractal models

    Broman, Erik I.; Jonasson, Johan; Tykesson, Johan
    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.

  4. Galton-Watson probability contraction

    Podder, Moumanti; Spencer, Joel
    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...

  5. Remarks on spectral gaps on the Riemannian path space

    Fang, Shizan; Wu, Bo
    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.

  6. Convergence of complex martingales in the branching random walk: the boundary

    Kolesko, Konrad; Meiners, Matthias
    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...

  7. Some connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisets

    Pinsky, Ross G.
    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.

  8. Self-averaging sequences which fail to converge

    Cator, Eric; Don, Henk
    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.

  9. Erratum: Optimal linear drift for the speed of convergence of an hypoelliptic diffusion

    Guillin, Arnaud; Monmarché, Pierre
    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.

  10. First passage percolation on a hyperbolic graph admits bi-infinite geodesics

    Benjamini, Itai; Tessera, Romain
    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...

  11. Indicable groups and $p_c<1$

    Raoufi, Aran; Yadin, Ariel
    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.

  12. The set of connective constants of Cayley graphs contains a Cantor space

    Martineau, Sébastien
    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.

  13. Product space for two processes with independent increments under nonlinear expectations

    Gao, Qiang; Hu, Mingshang; Ji, Xiaojun; Liu, Guomin
    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.

  14. Necessary and sufficient conditions for the $r$-excessive local martingales to be martingales

    Urusov, Mikhail; Zervos, Mihail
    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...

  15. Stable limit theorem for $U$-statistic processes indexed by a random walk

    Franke, Brice; Pène, Françoise; Wendler, Martin
    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

  16. Yet another condition for absence of collisions for competing Brownian particles

    Ichiba, Tomoyuki; Sarantsev, Andrey
    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.

  17. On recurrence and transience of multivariate near-critical stochastic processes

    Kersting, Götz
    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].

  18. Universal large deviations for Kac polynomials

    Butez, Raphaël; Zeitouni, Ofer
    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.

  19. Recurrence of multiply-ended planar triangulations

    Gurel-Gurevich, Ori; Nachmias, Asaf; Souto, Juan
    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 $0$.

  20. Recurrence and transience properties of multi-dimensional diffusion processes in selfsimilar and semi-selfsimilar random environments

    Kusuoka, Seiichiro; Takahashi, Hiroshi; Tamura, Yozo
    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...

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