The rate of convergence for the renewal theorem in $\mathbb{R}^d$
Résumé
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 hölder-continuous functions on the sphere.
Origine : Fichiers produits par l'(les) auteur(s)
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