Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case

Sara Brofferio, Dariusz Buraczewski, and Ewa Damek

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Abstract

We consider the autoregressive model on ℝd defined by the stochastic recursion Xn = AnXn−1 + Bn, where {(Bn, An)} are i.i.d. random variables valued in ℝd × ℝ+. The critical case, when $\mathbb{E}[\log A_{1}]=0$, was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measure ν for the Markov chain {Xn}. In the present paper we prove that the weak limit of properly dilated measure ν exists and defines a homogeneous measure on ℝd ∖ {0}.

Résumé

Nous considérons le modèle autorégressif sur ℝd défini par récurrence par l’équation stochastique Xn = AnXn−1 + Bn, où {(Bn, An)} sont des variables aléatoires à valeurs dans ℝd × ℝ+, indépendantes et de même loi. Le cas critique, c’est-à-dire lorsque $\mathbb{E}[\logA_{1}]=0$, a été étudié par Babillot, Bougerol et Elie, qui ont montré qu’il existe une et une seule mesure de Radon ν invariante pour la chaîne de Markov {Xn}. Dans ce papier nous démontrons que la mesure ν, convenablement dilatée, converge faiblement vers une mesure homogène sur ℝd ∖ {0}.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 2 (2012), 377-395.

Dates
First available in Project Euclid: 11 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1334148204

Digital Object Identifier
doi:10.1214/10-AIHP406

Mathematical Reviews number (MathSciNet)
MR2954260

Subjects
Primary: Primary 60J10 secondary 60B15 60G50: Sums of independent random variables; random walks

Keywords
Random walk Random coefficients autoregressive model Affine group Random equations Contractive system Regular variation

Citation

Brofferio, Sara; Buraczewski, Dariusz; Damek, Ewa. On the invariant measure of the random difference equation X n = A n X n −1 + B n in the critical case. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 2, 377--395. doi:10.1214/10-AIHP406. https://projecteuclid.org/euclid.aihp/1334148204


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