# The rate of convergence for the renewal theorem in $mathbb{R}^d$

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1 MODAL-X - Modélisation aléatoire de Paris X 2 IMB - Institut de Mathématiques de Bordeaux

Abstract : Let $ho$ be a borelian probability measure on $\mathrm{SL} d\mathbb{R}$. Consider the random walk $X n$ on $\mathbb{R}^d\setminus\{0\}$ defined by $ho$ : for any $x\in \mathbb{R}^d\setminus\{0\}$, we set $X 0 =x$ and $X {n+1} = g {n+1} X n$ where $g n$ is an iid sequence of $\mathrm{SL} d\mathbb{R}-$valued random variables of law $ho$. Guivarc-h and Raugi proved that under an assumption on the subgroup generated by the support of $ho$ strong irreducibility and proximality, this walk is transient.In particular, this proves that if $f$ is a compactly supported continuous function on $\mathbb{R}^d$, then the function $Gfx :=\mathbb{E} x \sum {n=0}^{+\infty} fX n$ is well defined for any $x\in \mathbb{R}^d \setminus\{0\}$.Guivarc-h and Le Page proved the renewal theorem in this situation : they study the possible limits of $Gf$ at $0$ and in this article, we study the rate of convergence in their renewal theorem.To do so, we consider the family of operators $Pit {t\in \mathbb{R}}$ defined for any continuous function $f$ on the sphere $\mathbb{S}^{d-1}$ and any $x\in \mathbb{S}^{d-1}$ by\Pit fx = \int {\mathrm{SL} d\mathbb{R}} e^{-it \ln \frac{ \|gx\|}{\|x\|}} f\left\frac{gx}{\|gx\|} ight \mathrm{d} hog\And we prove that, for some $L\in \mathbb{R}$ and any $t 0 \in \mathbb{R} +^\ast$,\\sup {\substack{t\in \mathbb{R}\\ |t| \geqslant t 0}} \frac{ 1 }{|t|^L} \left\| I d-Pit^{-1} ight\| \text{ is finite}\where the norm is taken in some space of hölder-continuous functions on the sphere.

Autor: Jean-Baptiste Boyer -

Fuente: https://hal.archives-ouvertes.fr/

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