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

Abstract

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 $Gf(x) :=\mathbb{E}_x \sum_{n=0}^{+\infty} f(X_n)$ 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 \[ P(it) f(x) = \int_{\mathrm{SL}_d(\mathbb{R})} e^{-it \ln \frac{ \|gx\|}{\|x\|}} f\left(\frac{gx}{\|gx\|}\right) \mathrm{d}\rho(g) \] And we prove that, for some $L\in \mathbb{R}$ and any $t_0 \in \mathbb{R}_+^\ast$, \[ \sup_{\substack{t\in \mathbb{R}\\ |t| \geqslant t_0}} \frac{ 1 }{|t|^L} \left\| (I_d-P(it))^{-1} \right\| \text{ is finite} \] where the norm is taken in some space of h\"older-continuous functions on the sphere.