On the invariant measure of the random difference equation X n =A n X n-1 +B n in the critical case
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 2, p. 377-395

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[logA 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}.

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}.

DOI : https://doi.org/10.1214/10-AIHP406
Classification:  Primary 60J10,  secondary,  60B15,  60G50
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},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {2},
     year = {2012},
     pages = {377-395},
     doi = {10.1214/10-AIHP406},
     zbl = {1259.60077},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIHPB_2012__48_2_377_0/

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