A note on recurrent random walks

We consider a recurrent RWRE $(X_n)_{n \in \mathbb{N}_0}$ on $\mathbb{Z}$ and investigate the quenched return probabilities of the RWRE to the origin for which we state results on their decay in terms of summability. Additionally, we give some examples for recurrent RWRE with a multidimensional state space which give reason for the part "strong recurrence" in the title of this paper when we compare the behaviour of RWRE with the behaviour of the symmetric random walk on $\mat...

In this article we refine well-known results concerning the fluctuations of one-dimensional random walks. More precisely, if $(S_n)_{n \geq 0}$ is a random walk starting from 0 and $r\geq 0$, we obtain the precise asymptotic behavior as $n\to\infty$ of $\mathbb P[\tau^{>r}=n, S_n\in K]$ and $\mathbb P[\tau^{>r}>n, S_n\in K]$, where $\tau^{>r}$ is the first time that the random walk reaches the set $]r,\infty[$, and $K$ is a compact set. Our assumptions on the jumps of the ran...

We consider a walker that at each step keeps the same direction with a probabilitythat depends on the time already spent in the direction the walker is currently moving. In this paper, we study some asymptotic properties of this persistent random walk and give the conditions of recurrence or transience in terms of "transition" probabilities to keep on the same direction or to change, without assuming that the latter admits any stationary probability. Examples are exhibited wh...

We study recurrence properties and the validity of the (weak) law of large numbers for (discrete time) processes which, in the simplest case, are obtained from simple symmetric random walk on $\Z$ by modifying the distribution of a step from a fresh point. If the process is denoted as $\{S_n\}_{n \ge 0}$, then the conditional distribution of $S_{n+1} - S_n$ given the past through time $n$ is the distribution of a simple random walk step, provided $S_n$ is at a point which has...

Let $\xi$ n , n $\in$ N be a sequence of i.i.d. random variables with values in Z. The associated random walk on Z is S(n) = $\xi$ 1 + $\times$ $\times$ $\times$ + $\xi$ n+1 and the corresponding "reflected walk" on N 0 is the Markov chain X(n), n $\in$ N, given by X(0) = x $\in$ N 0 and X(n + 1) = |X(n) + $\xi$ n+1 | for n $\ge$ 0. It is well know that the reflected walk (X(n)) n$\ge$0 is null-recurrent when the $\xi$ n are square integrable and centered. In this paper, we p...

In this note, we prove without using Fourier analysis that the symmetric square integrable random walks in $\Z^{2}$ are recurrent.

Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n - Y_{n+1}|$. In this note, we study recurrence of this process, extending a previous criterion. This is obtained by determining an invariant measure of the embedded process of reflections.

We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing symmetry the (sharp) inequality P(|X+Y| <= 1) < 2 P(|X-Y| <= 1) for independent identically distributed X,Y is used. In part II we shall discuss the connection of this result to 'polygonal recurrence' of higher-dimensional walks and some conj...

This paper has been withdrawn by the author due to a mistake in one of the main lemmas.

In the present paper we find necessary and sufficient conditions for recurrence of random walks on arbitrary subgroups of the group of rational numbers $\mathbb{Q}$.