Return Probabilities for the Reflected Random Walk on ${\mathbb{N}}_0$

Abstract

Let $(Y_n)$ be a sequence of i.i.d. $\mathbb{Z}$-valued random variables with law $\mu$. The reflected random walk $(X_n)$ is defined recursively by $X_0=x\in {\mathbb{N}}_0,X_{n+1}=|X_n+Y_{n+1}|$. Under mild hypotheses on the law $\mu$, it is proved that, for any $y\in {\mathbb{N}}_0$, as $n\to +\infty$, one gets ${\mathbb{P}}_x[X_n=y]\sim C_{x,y}{R}^{-n}{n}^{-3/2}$ when ${\sum }_{k\in \mathbb{Z}}k\mu (k)>0$ and ${\mathbb{P}}_x[X_n=y]\sim C_y{n}^{-1/2}$ when ${\sum }_{k\in \mathbb{Z}}k\mu (k)=0$, for some constants $R,C_{x,y}$ and $C_y>0$.

References in corpus (1)

• Renewal theory and computable convergence rates for geometrically ergodic Markov chains