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

Jean-Baptiste Boyer

Published 2016-03-23Version 1

Let $\rho$ 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 $\rho$ : 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 $\rho$. Guivarc'h and Raugi proved that under an assumption on the subgroup generated by the support of $\rho$ (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 $Gf(x) :=\mathbb{E}\_x \sum\_{k=0}^{+\infty} f(X\_n)$ 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 $(P(it))\_{t\in \mathbb{R}}$ defined for any continuous function $f$ on the sphere $\mathbb{S}^{d-1}$ and any $x\in \mathbb{S}^{d-1}$ by\[P(it) f(x) = \int\_{\mathrm{SL}\_d(\mathbb{R})} e^{-it \ln \frac{\|gx\|}{\|x\|}} f\left(\frac{gx}{\|gx\|}\right) \mathrm{d}\rho(g)\]We prove that, adding an exponential moment condition to the strong irreducibility and proximality condition, we have that for some $L\in \mathbb{R}$ and any $t\_0 \in \mathbb{R}\_+^\ast$,\[\sup\_{\substack{t\in \mathbb{R}\\ |t| \geqslant t\_0}} \frac{1}{|t|^L} \left\| (I\_d-P(it))^{-1} \right\| \text{is finite}\]where the norm is taken in some space of h\"older-continuous functions on the sphere.