Return probabilities for the reflected random walk on N_0

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.