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 , 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}.
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 , 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}.
Keywords: random walk, random coefficients autoregressive model, affine group, random equations, contractive system, regular variation
@article{AIHPB_2012__48_2_377_0, author = {Brofferio, Sara and Buraczewski, Dariusz and Damek, Ewa}, title = {On the invariant measure of the random difference equation $X_n=A_nX_{n-1}+B_n$ in the critical case}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {377--395}, publisher = {Gauthier-Villars}, volume = {48}, number = {2}, year = {2012}, doi = {10.1214/10-AIHP406}, zbl = {1259.60077}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/10-AIHP406/} }
TY - JOUR AU - Brofferio, Sara AU - Buraczewski, Dariusz AU - Damek, Ewa TI - On the invariant measure of the random difference equation $X_n=A_nX_{n-1}+B_n$ in the critical case JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 377 EP - 395 VL - 48 IS - 2 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/10-AIHP406/ DO - 10.1214/10-AIHP406 LA - en ID - AIHPB_2012__48_2_377_0 ER -
%0 Journal Article %A Brofferio, Sara %A Buraczewski, Dariusz %A Damek, Ewa %T On the invariant measure of the random difference equation $X_n=A_nX_{n-1}+B_n$ in the critical case %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 377-395 %V 48 %N 2 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/10-AIHP406/ %R 10.1214/10-AIHP406 %G en %F AIHPB_2012__48_2_377_0
Brofferio, Sara; Buraczewski, Dariusz; Damek, Ewa. On the invariant measure of the random difference equation $X_n=A_nX_{n-1}+B_n$ in the critical case. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 2, pp. 377-395. doi : 10.1214/10-AIHP406. http://archive.numdam.org/articles/10.1214/10-AIHP406/
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