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

hal.structure.identifierModélisation aléatoire de Paris X [MODAL'X]
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorBOYER, Jean-Baptiste
dc.date.accessioned2024-04-04T03:14:26Z
dc.date.available2024-04-04T03:14:26Z
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/194055
dc.description.abstractEnLet $\rho$ 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.
dc.language.isoen
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/
dc.title.enThe rate of convergence for the renewal theorem in $\mathbb{R}^d$
dc.typeDocument de travail - Pré-publication
dc.subject.halMathématiques [math]/Probabilités [math.PR]
dc.subject.halMathématiques [math]/Systèmes dynamiques [math.DS]
dc.identifier.arxiv1603.07214
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-01292084
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01292084v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=BOYER,%20Jean-Baptiste&rft.genre=preprint


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