Recurrent Random Walk Research Articles

Propriétés d'intersection des marches aléatoires

We study intersection properties of multi-dimensional random walks. LetX andY be two independent random walks with values in ℤd (d≦3), satisfying suitable moment assumptions, and letIn denote the number of common points to the paths ofX andY up to timen. The sequence (In), suitably normalized, is shown to converge in distribution towards the “intersection local time” of two independent Brownian motions. Results are applied to the proof of a central limit theorem for the range of a two-dimensional recurrent random walk, thus answering a question raised by N. C. Jain and W. E. Pruitt.

How small are the increments of the local time of a recurrent random walk?

How small are the increments of the local time of a recurrent random walk?

Extreme sojourns for random walks and birth-and-death processes

Let (X n) be a recurrent random walk on the nonnegative integers such that, at each stepX n increases or decreases by 1. The transition probabilities depend on the current state. For any starting point put visits to state n before the first visit to 0. Similarly, let (X t) be a recurrent birth-and-death process on the non-negative integers, and put L n = time spent in state n before the first passage time to 0. Exact asymptotic formulas are obtained for the tails of the distributions of M nand Ln , for under various conditions on the transition probabilities and the birth and death rates of the respective processes. For certain numbers (Bn ) associated with these processes, define the long term sojourns visits to n up to time [Bn ], and sojourn time in n up to time Bn . Under the additional hypothesis of ergodicity, it is shown what , suitably normalized, have limiting compound Poisson distributions, where the compounding distributions are the geometric and exponential, respectively

Marches aléatoires récurrentes totalement dissymétriques et domaine d⊃attraction des lois stables

The aim of this article is to answer some question raised by A. Brunei and D. Revuz concerning the Martin boundary for recurrent random walks on groups.Let G be a group and a the recurrent potential measure of a random walk on We characterise those random walks for which lim infx , x a{x) is finite, showing that this property is related to the renewal theory. As an example,we prove that, on Z, the walks in the domain of normal attraction of any centered and maximally asymmetric stable walk with parameter ae]l,2[ share this property

The Asymptotic Behavior of Local Times of Recurrent Random Walks with Infinite Variance

The Asymptotic Behavior of Local Times of Recurrent Random Walks with Infinite Variance

Asymptotic Behavior of the Local Time of a Recurrent Random Walk

Let $(S_j)$ be a lattice random walk, i.e. $S_j = X_1 + \cdots + X_j$ where $X_1, X_2, \cdots$ are independent random variables with values in the integer lattice and common nondegenerate distribution $F$, and let $L_n(x) = \sum^{n - 1}_{j = 0} 1_{\{x\}} (S_j)$, the local time of the random walk at $x$ before time $n$. Define $G(x) = P\{|X_1| > x\}, K(x) = x^{-2} \int_{|y| \leq x}y^2 dF (y), Q(x) = G(x) + K(x)$ for $x > 0. Q$ is continuous, strictly decreasing for large $x$, and tends to zero. Thus $a_y$ may be defined by $Q(a_y) = y^{-1}$ for large $y$ and then we let $c_n = a_{n/\log\log n}$. The basic assumption is that $\lim \sup_{x \rightarrow \infty} G(x)/K(x) < 1$ and $EX_1 = 0$. We prove that there exist positive constants $\theta_1, \theta_2$ such that $\lim \sup_{n \rightarrow \infty} c_nn^{-1}L_n(x) = \theta_1$ a.s. for all $x, \lim \sup_{n \rightarrow \infty} \sup_xc_nn^{-1}L_n(x) = \theta_2$ a.s. Furthermore $\lim_{\delta \rightarrow 0} \lim \sup_{n \rightarrow \infty} \sup_{|x - y| \leq \delta c_n}c_nn^{-1}|L_n(x) - L_n(y)| = 0 a.s.$ One of the main tools is an estimate for the "absolute potential kernel": $\sum^\infty_{n = 0}| P\{S_n = z\} - P\{S_n = z + x\}| \leq C(|x|Q(|x|))^{-1}$ assuming strong aperiodicity.

The narrowest tube of a recurrent random walk

Let X1, X2, ... be i.i.d. random variables with P(X1=+1)=P(X1 =−1)=1/2. Put S0=0, Sn=X1+...+Xn (n≧1). Our aim is to investigate the a.s. behavior of $$U(a_N ,N) = \mathop {\min }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } j\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } N - a_N } \mathop {\max }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } a_N } \left| {S_{j + i} - S_j } \right|$$ and $$V(a_N ,N) = \mathop {\min }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } j\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } N - a_N } \mathop {\max }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } a_N } \left| {S_{j + i} } \right|$$ . It is shown that for aN=[c log N] both U(aN, N) and V(aN, N) are a.s. constant for large N, except for certain values of c, when U and V can take two values for large N. The result is extended for recurrent random walk too.

Limit Theorems for Sums of Independent Random Variables Defined on a Recurrent Random Walk

Let {Xi}i=∞ -∞, {ξi}i=1∞ be two independent sequences of randoms variables, where the ξi are identically distributed and assume integer values. Let In the paper the question of the asymptotic behavior as n → ∞ of the quantity is considered. It is shown that the distribution of Wn converges to the distribution of the normal law and that the estimate of the rate of convergence has the same order as the classical estimate of Berry-Esseen.

An invariance principle for the local time of a recurrent random walk

Let (S j ) be a lattice random walk, i.e. S j =X 1 +...+X j , where X 1,X 2,... are independent random variables with values in the integer lattice ℤ and common distribution F, and let \(L_n (\omega ,k) = \sum\limits_{j = 0}^{n - 1} {\chi _{\{ k\} } } (S_j (\omega ))\), the local time of the random walk at k before time n. Suppose EX 1=0 and F is in the domain of attraction of a stable law G of index α> 1, i.e. there exists a sequence a(n) (necessarily of the form n 1αl(n), where l is slowly varying) such that S n /a(n)→ G. Define \(g_n (\omega ,u) = \frac{{c(n)}}{n}L_n (\omega ,[uc(n)])\), where c(n)=a(n/log log n) and [x] = greatest integer ≦ x. Then we identify the limit set of {g n (ω, ·)∶ n≧1} almost surely with a nonrandom set in terms of the I-functional of Donsker and Varadhan.The limit set is the one that Donsker and Varadhan obtain for the corresponding problem for a stable process. Several corollaries are then derived from this invariance principle which describe the asymptotic behavior of L n (ω, ·) as n→∞.

On the Asymptotic Behavior of Local Times of Recurrent Random Walks with Finite Variance

On the Asymptotic Behavior of Local Times of Recurrent Random Walks with Finite Variance

On Recurrent Random Walks on Abelian Semigroups

On Recurrent Random Walks on Abelian Semigroups

On recurrent random walks on abelian semigroups

We consider a random walk Z n {Z_n} on the locally compact second countable abelian semigroup S with unit e which is assumed to be a recurrent point for Z n {Z_n} . Then S is the disjoint union of a topologically simple abelian semigroup G and a null-set A of first category. Under some additional conditions G is a topological group.

On the Range of Recurrent Markov Chains

Let $R_n$ be the number of distinct elements among $X_0, X_1,\cdots, X_n$, where $\{X_n\}$ is an irreducible recurrent Markov chain. It is shown that, under an appropriate condition, $n^{-1}R_n \rightarrow 0$ a.s. $(P_a)$ where $a$ is any state and $P_a$ is conditional probability measure given $X_0 = a$. We prove that any recurrent random walk satisfies our condition, so that the result contains the well-known random walk case. We also give an example of an irreducible recurrent chain for which the result fails to hold.

A Ratio Limit Theorem for Subterminal Times

Consider a recurrent random walk $\{X_n\}$ with state space $S \subseteqq R^d (d \leqq 2)$. A stopping time $T$ is called subterminal if it satisfies a technical condition which essentially states that it is the first time a path possesses some property which does not depend on how long the process has been running. Suppose $T$ is a subterminal time which can occur only when $\{X_n\}$ is in a bounded set; then under an additional assumption a ratio limit theorem (as $n \rightarrow \infty$) is obtained for $P(T > n \mid X_0 = x) (x\in S)$. The theorem applies in particular to the hitting time of a bounded set with nonempty interior in the general case, and to the hitting time of a bounded set with nonzero Haar measure in the nonsingular case.

An Invariance Principle for Random Walk Conditioned by a Late Return to Zero

Let $\{S_n: n \geqq 0\}$ denote the recurrent random walk formed by the partial sums of i.i.d. integer-valued random variables with zero mean and finite variance. Let $T = \min \{n \geqq 1: S_n = 0\}$. Our main result is an invariance principle for the random walk conditioned by the event $\lbrack T = n\rbrack$. The limiting process is identified as a Brownian excursion on [0, 1].

Recurrent random walks on certain classes of groups

This article is divided into two parts: in the first we give some results about renewal and normality of a recurrent random walk (r.w.) on an abelian group, without the Harris hypothesis, which will extend the theorems of S.C. Port and C.J. Stone ([8]) to a larger class of functions. They are stated in the Theorems 1.14 and 1.15. The technique will be to approximate the recurrent r.w. by a Harris recurrent r.w., for which the recent results of A. Brunel and D. Revuz ([2–4]) hold.

A Conditional Local Limit Theorem for Recurrent Random Walk

Let $S_n, n = 1, 2, 3, \cdots$ denote the recurrent random walk formed by the partial sums of i.i.d. lattice random variables with mean zero and finite variance. Let $T_{\{x\}} = \min \lbrack n \geqq 1 \mid S_n = x \rbrack$ with $T \equiv T_{\{0\}}$. We obtain a local limit theorem for the random walk conditioned by the event $\lbrack T > n \rbrack$. This result is applied then to obtain an approximation for $P\lbrack T_{\{x\}} = n \rbrack$ and the asymptotic distribution of $T_{\{x\}}$ as $x$ approaches infinity.

SLLNs and CLTs for Infinite Particle Systems

We consider initial point processes $A_0$ on $Z^d$ where $A_0(x), x \in Z^d$ are independent and satisfy certain technical conditions. The particles initially present are assumed to be translated independently according to recurrent random walks. Various limit theorems are then proved involving $S_n(B)$--the total occupation time of $\mathbf{B}$ by time $n$, and $L_n(\mathbf{B})$--the number of distinct particles in $\mathscr{B}$ by time $n$.

Recurrent Random Walks on Countable Abelian Groups of Rank 2

Let $r_n$ be the probability that a recurrent random walk on a countable Abelian group fails to return to the origin in the first $n$ steps. For two-dimensional walk, Kesten and Spitzer have shown that $r_n$ is slowly varying. I.e. $\lim_{n\rightarrow\infty} r_{2n}/r_n = 1$. We strengthen this result and show that for any countable Abelian group of rank 2, $r_n$ is super slowly varying in the sense that $\lim_{n\rightarrow\infty} r_{\lbrack nr_n \rbrack}/r_n = 1$. We use the superslow variation of $r_n$ to obtain the limit law for the number of returns to the origin for all recurrent random walks on these groups.

Recurrent random walk of an infinite particle system

Let $p(x,y)$ be the transition function for a symmetric irreducible recurrent Markov chain on a countable set $S$. Let ${\eta _t}$ be the infinite particle system on $S$ moving according to simple exclusion interaction with the one particle motion determined by $p$. Assume that $p$ is such that any two particles moving independently on $S$ will sooner or later meet. Then it is shown that every invariant measure for ${\eta _t}$ is a convex combination of Bernoulli product measures ${\mu _\alpha }$ on ${\{ 0,1\} ^s}$ with density $0 \leqslant \alpha = \mu [\eta (x) = 1] \leqslant 1$. Ergodic theorems are proved concerning the convergence of the system to one of the ${\mu _\alpha }$.