On the invariant measure of the random difference equation $X_n=A_n{X}_{n-1}+B_n$ in the critical case

Brofferio, Sara; Buraczewski, Dariusz; Damek, Ewa

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 377-395.

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 $𝔼{[}_1]=0$ , 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}.

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 $𝔼[log{A}_1]=0$ , 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}.

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