# Return probabilities for the reflected random walk on N_0

R. Essifi, M. Peigné

3 Citations (Scopus)
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 $$X_0=x \in \mathbb{N }_0, X_{n+1}=\vert X_n+Y_{n+1}\vert$$. Under mild hypotheses on the law $$\mu$$, it is proved that, for any $$y \in \mathbb{N }_0$$, as $$n \rightarrow +\infty$$, 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 $$C_y >0$$. Keywords: Random walks Local limit theorem Generating function Wiener-Hopf factorization An erratum to this article can be found at http://¿dx.¿doi.¿org/¿10.¿1007/¿s10959-014-0568-6.