Search ResultsShowing 1-9 of 9
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arXiv:2009.13816 (Published 2020-09-29)
Maximal local time of randomly biased random walks on a Galton-Watson tree
Categories: math.PRWe consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson tree with countably infinitely many types. When the walk is the diffusive or sub-diffusive, by studying the maximal type of this Galton-Watson tree, we establish the asymptotic behaviour of the largest local times of this walk during n excursions, under the annealed law.
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arXiv:1805.03971 (Published 2018-05-10)
Potential function, ladder variables and absorption probabilities of a recurrent random walk on $\mathbb{Z}$ with infinite variance
Comments: 24 pagesCategories: math.PRWe consider a recurrent random walk of i.i.d. increments on the one dimensional integer lattice and obtain a certain relation between the ladder height distribution and the potential function, $a(x)$, of the walk. Applying it we derive an asymptotic estimate of $a(x)$ and thereby a criterion for $a$ to be bounded on a half line. We also apply it to a classical two-sided exit problem and show that if the expectation of the ladder height is finite, then Spitzer's condition is necessary and sufficient for the probabilities of exiting a long interval $[-M,N]$ at, say, the right end to converge whenever $M/N$ tends to a positive constant.
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arXiv:1711.10202 (Published 2017-11-28)
Empirical processes for recurrent and transient random walks in random scenery
Categories: math.PRIn this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where $(\xi_x, x\in\mathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{n\in\mathbb N}$ is a random walk evolving in $\mathbb{Z}^d$, independent of the $\xi$'s. In Wendler (2016), the case where $(S_n)_{n\in\mathbb N}$ is a recurrent random walk in $\mathbb{Z}$ such that $(n^{-\frac 1\alpha}S_n)_{n\geq 1}$ converges in distribution to a stable distribution of index $\alpha$, with $\alpha\in(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{n\in\mathbb N}$ is either: a) a transient random walk in $\mathbb{Z}^d$, b) a recurrent random walk in $\mathbb{Z}^d$ such that $(n^{-\frac 1d}S_n)_{n\geq 1}$ converges in distribution to a stable distribution of index $d\in\{1,2\}$.
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arXiv:1612.00917 (Published 2016-12-03)
Average Entropy of the Ranges for Simple Random Walks on Discrete Groups
Comments: 28 pagesCategories: math.PRInspired by Benjamini et al (Ann. Inst. H. Poincar\'e Probab. Stat. 2010) and Windisch (Electron. J. Probab. 2010), we consider the entropy of the random walk range formed by a simple random walk on a discrete group. It is shown in this setting the existence of a quantity which we call the average entropy of the range. The extreme values of this quantity are also discussed. An important consequence is that, the tail $\sigma$-algebra formed by the $n$-step ranges is always trivial for a recurrent random walk.
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arXiv:1511.02810 (Published 2015-11-09)
Exponentials and $R$-recurrent random walks on groups
Categories: math.PROn a locally compact group $E$ with countable base, we consider a random walk $X$ that has a unique (up to a positive factor) $r$-invariant measure for some $r>0$. Under some weak conditions on the measure, there is a unique continuous exponential on $E$ naturally associated with $X$. It follows that there exists an $R$-recurrent random walk in the sense of Tweedie on $E$ if and only if $E$ is a recurrent group and there exists a Harris random walk on~$E$.
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arXiv:0710.5854 (Published 2007-10-31)
Lingering random walks in random environment on a strip
Categories: math.PRKeywords: lingering random walks, random environment, asymptotic behaviour, recurrent random walk, central limit theoremTags: journal articleWe consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the "(log t)-squared" asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the $(\log t)^{2}$ asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem.
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arXiv:math/0610056 (Published 2006-10-02)
A note on recurrent random walks
Comments: 8 pagesJournal: Statistics and Probability Letters 76 (2006) 1025-1031Categories: math.PRSubjects: 60F99Keywords: recurrent random walk, increasing sequences, symmetric random walks, finite accumulation point, simpler proofTags: journal articleFor any recurrent random walk (S_n)_{n>0} on R, there are increasing sequences (g_n)_{n>0} converging to infinity for which (g_n S_n)_{n>0} has at least one finite accumulation point. For one class of random walks, we give a criterion on (g_n)_{n>0} and the distribution of S_1 determining the set of accumulation points for (g_n S_n)_{n>0}. This extends, with a simpler proof, a result of K.L. Chung and P. Erdos. Finally, for recurrent, symmetric random walks, we give a criterion characterizing the increasing sequences (g_n)_{n>0} of positive numbers for which liminf g_n|S_n|=0.
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arXiv:math/0608036 (Published 2006-08-01)
Slow movement of random walk in random environment on a regular tree
Categories: math.PRWe consider a recurrent random walk in random environment on a regular tree. Under suitable general assumptions upon the distribution of the environment, we show that the walk exhibits an unusual slow movement: the order of magnitude of the walk in the first $n$ steps is $(\log n)^3$.
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arXiv:math/0603363 (Published 2006-03-15)
A subdiffusive behaviour of recurrent random walk in random environment on a regular tree
Comments: 29 pages with 1 figure. Its preliminary version was put in the following web site: http://www.math.univ-paris13.fr/prepub/pp2005/pp2005-28.htmlCategories: math.PRWe are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk $(X\_n)$ in random environment on a regular tree, which is closely related to Mandelbrot [13]'s multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent $\nu\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$ behaves asymptotically like $n^{\nu}$. The value of $\nu$ is explicitly formulated in terms of the distribution of the environment.