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}.
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}.
Mots clés : 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://www.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://www.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://www.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, Tome 48 (2012) no. 2, pp. 377-395. doi : 10.1214/10-AIHP406. http://www.numdam.org/articles/10.1214/10-AIHP406/
[1] The random difference equation Xn = AnXn−1 + Bn in the critical case. Ann. Probab. 25 (1997) 478-493. | Zbl
, and .[2] Contractive stochastic dynamical systems. Unpublished manuscript, Ludwig-Maximilians-Universität München, 1999.
.[3] How a centred random walk on the affine group goes to infinity. Ann. Inst. Henri Poincaré Probab. Statist. 39 (2003) 371-384. | Numdam | Zbl
.[4] On invariant measures of stochastic recursions in a critical case. Ann. Appl. Probab. 17 (2007) 1245-1272. | Zbl
.[5] On tails of fixed points of the smoothing transform in the boundary case. Stochastic Process. Appl. 119 (2009) 3955-3961. | Zbl
.[6] Asymptotic behavior of Poisson kernels NA group. Comm. Partial Differential Equations 31 (2006) 1547-1589. | Zbl
, and .[7] Random walks on the affine group of local fields and of homogeneous trees. Ann. Inst. Fourier (Grenoble) 44 (1994) 1243-1288. | Numdam | Zbl
, and .[8] Asymptotic behavior of the invariant measure for a diffusion related to a NA group. Colloq. Math. 104 (2006) 285-309. | Zbl
and .[9] Sur l'équation de convolution μ = μ * σ. C. R. Acad. Sci. Paris 250 (1960) 799-801 (in French). | Zbl
and .[10] Comportement asymptotique du noyau potentiel sur les groupes de Lie. Ann. Sci. École Norm. Sup. (4) 15 (1982) 257-364. | Numdam | Zbl
.[11] An Introduction to Probability Theory and Its Application, Vol. II. Wiley, New York, 1966. | MR | Zbl
.[12] Hardy Spaces on Homogeneous Groups. Princeton Univ. Press, Princeton, NJ, 1982. | MR | Zbl
and .[13] Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 (1991) 126-166. | MR | Zbl
.[14] Limit theorem for products of random linear transformations of the line. Litovsk. Mat. Sb. 15 (1975) 61-77, 241 (in Russian). | Zbl
.[15] On a limit distribution for a random walk on lines. Litovsk. Mat. Sb. 15 (1975) 79-91, 243 (in Russian). | MR | Zbl
.[16] Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973) 207-248. | MR | Zbl
.[17] A local limit theorem on the semi-direct product of R∗+ and Rd. Ann. Inst. Henri Poincaré Probab. Statist. 33 (1997) 223-252. | Numdam | MR | Zbl
and .[18] Hitting time and hitting places for non-lattice recurrent random walks. J. Math. Mech. 17 (1967) 35-57. | MR | Zbl
and .[19] Potential theory of random walks on Abelian groups. Acta Math. 122 (1969) 19-114. | MR | Zbl
and .Cité par Sources :