# 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 $X_0=x \in \mathbb N_0, X_{n+1}=\vert X_n+Y_{n+1}\vert$. Under mild hypotheses on the law $\mu$, it is proved that, for any $y \in \mathbb N_0$, as $n \to +\infty$, 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 $C_y >0$.
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https://hal.archives-ouvertes.fr/hal-00711077
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Submitted on : Friday, June 29, 2012 - 8:16:23 AM
Last modification on : Friday, February 19, 2021 - 4:10:02 PM
Long-term archiving on: : Sunday, September 30, 2012 - 2:20:39 AM

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### Identifiers

• HAL Id : hal-00711077, version 1
• ARXIV : 1206.6953

### Citation

Rim Essifi, Marc Peigné. Return Probabilities for the Reflected Random Walk on $\mathbb N_0$. 2012. ⟨hal-00711077⟩

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