arXiv subjects

Arvind Singh

Publications and source records attributed to Arvind Singh.

40 records · Page 3Linked to original sources

arXiv2007

Rate of growth of a transient cookie random walk

We consider a one-dimensional transient cookie random walk. It is known from a previous paper that a cookie random walk $(X_n)$ has positive or zero speed according to some positive parameter $α>1$ or $\le 1$. In this article, we give the exact rate of growth of $(X_n)$ in the zero speed regime, namely: for $0<α<1$, $X_n/n^{\frac{α+1}{2}}$ converges in law to a Mittag-Leffler distribution whereas for $α=1$, $X_n(\log n)/n$ converges in probability to some positive constant.

math.PR↗

arXiv2006

A slow transient diffusion in a drifted stable potential

We consider a diffusion process $X$ in a random potential $\V$ of the form $\V_x = §_x -δx$ where $δ$ is a positive drift and $§$ is a strictly stable process of index $α\in (1,2)$ with positive jumps. Then the diffusion is transient and $X_t / \log^αt$ converges in law towards an exponential distribution. This behaviour contrasts with the case where $\V$ is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as "slow" as in the recurrent setting.

math.PR↗

arXiv2006

On the speed of a cookie random walk

We consider the model of the one-dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite. We give a criterion to decide whether the limiting speed of the walk is non-zero. In particular, we show that a positive speed may be obtained for just 3 cookies per site. We also prove a result on the continuity of the speed with respect to the initial cookie distribution.

math.PR↗

arXiv2005

Limiting behavior of a diffusion in an asymptotically stable environment

Let $V$ be a two sided random walk and let $X$ denote a real valued diffusion process with generator ${1/2}e^{V([x])}\frac{d}{dx}(e^{-V([x])}\frac{d}{dx})$. This process is known to be the continuous equivalent of the one dimensional random walk in random environment with potential $V$. Hu and Shi (1997) described the Lévy classes of $X$ in the case where $V$ behaves approximately like a Brownian motion. In this paper, based on some fine results on the fluctuations of random walks and stable processes, we obtain an accurate image of the almost sure limiting behavior of $X$ when $V$ behaves asymptotically like a stable process. These results also apply for the corresponding random walk in random environment.

math.PR↗