A note on recurrent random walks

In this paper we prove that under certain assumptions the transient random walk in random environment with bounded jumps (in $\mathbb{Z}$) grows much slower than the speed $n$. Precisely, there is $0<s<1$, such that although $X_n\rto$ we have $\frac{X_n}{n^{s'}}\rightarrow 0$ for $0<s<s'$ almost surely.

Let $F$ be a distribution function on the integer lattice $\mathbb{Z}$ and $S=(S_n)$ the random walk with step distribution $F$. Suppose $S$ is oscillatory and denote by $U_{\rm a}(x)$ and $u_{\rm a}(x)$ the renewal function and sequence, respectively, of the strictly ascending ladder height process associated with $S$. Putting $A(x) =\int_0^x [1-F(t)-F(-t)] dt$, $H(x)=1-F(x)+F(-x)$ we suppose $$A(x)/\big(xH(x)\big) \to -\infty \quad (x\to\infty).$$ Under some additional re...

We consider the sums $S_n=\xi_1+\cdots+\xi_n$ of independent identically distributed random variables. We do not assume that the $\xi$'s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability ${\bf P}\{M>x\}$ as $x\to\infty$, provided that $M=\sup\{S_n,\ n\ge1\}$ is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some suf...

In this paper, we consider the $(1,R)$ state-dependent reflecting random walk (RW) on the half line, allowing the size of jumps to the right at maximal $R$ and to the left only 1. We provide an explicit criterion for positive recurrence and the explicit expression of the stationary distribution based on the intrinsic branching structure within the walk. As an application, we obtain the tail asymptotic of the stationary distribution in the "near critical" situation.

In this paper, we penalised the standard random walk by several functions of its maximum. The aim is to show that in spite of very close penalisation functions, under the new probabilities, the canonical process behaves very differently.

This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They...

We consider the recurrence and transience problem for a time-homogeneous Markov chain on the real line with transition kernel $p(x,\mathrm{d}y)=f_x(y-x)\,\mathrm{d}y$, where the density functions $f_x(y)$, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, where $\alpha(x)\in(0,2)$. In this paper, under a uniformity condition on the density functions $f_x(y)$ and an additional mild drift condition, we prove that when $\lim\inf_{|x|\longrightarrow\infty}\alph...

Let $(S_n)_{n \geq 0}$ be a transient random walk in the domain of attraction of a stable law and let $(\xi(s))_{s \in \mathbb{Z}}$ be a stationary sequence of random variables. In a previous work, under conditions of type $D(u_n)$ and $D'(u_n)$, we established a limit theorem for the maximum of the first $n$ terms of the sequence $(\xi(S_n))_{n\geq 0}$ as $n$ goes to infinity. In this paper we show that, under the same conditions and under a suitable scaling, the point proce...

In this expository note, we give a simple proof that a gambler repeating a game with positive expected value never goes broke with a positive probability. This does not immediately follow from the strong law of large numbers or other basic facts on random walks. Using this result, we provide an elementary proof of the strong law of large numbers. The ideas of the proofs come from the maximal ergodic theorem and Birkhoff's ergodic theorem.

In this short note we give various near optimal characterizations of random walks over finite Abelian groups with large maximum discrepancy from the uniform measure. We also provide several interesting connections to existing results in the literature.