Related papers: On a property of the simple random walk on $\mathb…
We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes…
Associated to a random walk on $\mathbb{Z}$ and a positive integer $n$, there is a return probability of the random walk returning to the origin after $n$ steps. An interesting question is when the set of return probabilities uniquely…
Random walk is one of the most classical and well-studied model in probability theory. For two correlated random walks on lattice, every step of the random walks has only two states, moving in the same direction or moving in the opposite…
When it comes to random walk on the integers $\mathbb{Z}$, the arguably first step of generalization beyond simple random walk is the class of one-sidedly continuous random walk, where the stepsize in only one direction is bounded by 1.…
The classic arcsine law for the number $N_{n}^{>}:=n^{-1}\sum_{k=1}^{n}\mathbf{1}_{\{S_{k}>0\}}$ of positive terms, as $n\to\infty$, in an ordinary random walk $(S_{n})_{n\ge 0}$ is extended to the case when this random walk is governed by…
Simple random walks on a partially directed version of $\mathbb{Z}^2$ are considered. More precisely, vertical edges between neighbouring vertices of $\mathbb{Z}^2$ can be traversed in both directions (they are undirected) while horizontal…
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…
In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density $\rho \in (0,\infty)$.…
A classical result for the simple symmetric random walk with $2n$ steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and…
In this paper, we investigate the distribution of $k$-free numbers in a class of $\alpha$-random walks on the integer lattice $\mathbb{Z}$. In these walks, the walker starts from a non-negative integer $r$ and moves to the right by $a$…
Consider two random walks on $\mathbb{Z}$. The transition probabilities of each walk is dependent on trajectory of the other walker i.e. a drift $p>1/2$ is obtained in a position the other walker visited twice or more. This simple model has…
We consider a random walk on integers where at the first visits to a site the walker gets a positive drift, but where after a certain number of visits the walker gets a negative drift. We prove that the walker is almost surely transient to…
Let $(S_k)_{k\ge 1}$ be the classical Bernoulli random walk on the integer line with jump parameters $p\in(0,1)$ and $q=1-p$. The probability distribution of the sojourn time of the walk in the set of non-negative integers up to a fixed…
To what extent is the underlying distribution of a finitely supported unbiased random walk on $\mathbb{Z}$ determined by the sequence of times at which the walk returns to the origin? The main result of this paper is that, in various…
We present an easy proof of Polya's theorem on random walks: with the probability one a random walk on the two-dimensional lattice returns to the starting point.
Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range,…
The Sparre-Andersen theorem is a remarkable result in one-dimensional random walk theory concerning the universality of the ubiquitous first-passage-time distribution. It states that the probability distribution $\rho_n$ of the number of…
Random walks, and in particular, their first passage times, are ubiquitous in nature. Using direct enumeration of paths, we find the first return time distribution of a 1D random walker, which is a heavy-tailed distribution with infinite…
We consider random walk on a finite group $G$ as follows. We can consider $G$ as a group of substitutions. Randomly (i.e. with probability $U(g)=|G|^{-1}$ ) we choose a substitution $g \in G$ and execute it twice in a row, i.e. execute a…
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…