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), p. 377-395 / Harvested from Numdam
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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[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}.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/10-AIHP406
Classification:  Primary 60J10,  secondary,  60B15,  60G50 
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     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},
     volume = {48},
     year = {2012},
     pages = {377-395},
     doi = {10.1214/10-AIHP406},
     zbl = {1259.60077},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_2_377_0}
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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) pp. 377-395. doi : 10.1214/10-AIHP406. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_2_377_0/

[1] M. Babillot, P. Bougerol and L. Élie. The random difference equation Xn = AnXn−1 + Bn in the critical case. Ann. Probab. 25 (1997) 478-493. | Zbl 0873.60045

[2] M. Benda. Contractive stochastic dynamical systems. Unpublished manuscript, Ludwig-Maximilians-Universität München, 1999.

[3] S. Brofferio. How a centred random walk on the affine group goes to infinity. Ann. Inst. Henri Poincaré Probab. Statist. 39 (2003) 371-384. | Numdam | Zbl 1016.60006

[4] D. Buraczewski. On invariant measures of stochastic recursions in a critical case. Ann. Appl. Probab. 17 (2007) 1245-1272. | Zbl 1151.60034

[5] D. Buraczewski. On tails of fixed points of the smoothing transform in the boundary case. Stochastic Process. Appl. 119 (2009) 3955-3961. | Zbl 1183.60031

[6] D. Buraczewski, E. Damek and A. Hulanicki. Asymptotic behavior of Poisson kernels NA group. Comm. Partial Differential Equations 31 (2006) 1547-1589. | Zbl 1109.22005

[7] D. I. Cartwright, V. A. Kaimanovich and W. Woess. Random walks on the affine group of local fields and of homogeneous trees. Ann. Inst. Fourier (Grenoble) 44 (1994) 1243-1288. | Numdam | Zbl 0809.60010

[8] E. Damek and A. Hulanicki. Asymptotic behavior of the invariant measure for a diffusion related to a NA group. Colloq. Math. 104 (2006) 285-309. | Zbl 1087.22006

[9] G. Choquet and J. Deny. Sur l'équation de convolution μ = μ * σ. C. R. Acad. Sci. Paris 250 (1960) 799-801 (in French). | Zbl 0093.12802

[10] L. Élie. Comportement asymptotique du noyau potentiel sur les groupes de Lie. Ann. Sci. École Norm. Sup. (4) 15 (1982) 257-364. | Numdam | Zbl 0509.60070

[11] W. Feller. An Introduction to Probability Theory and Its Application, Vol. II. Wiley, New York, 1966. | MR 210154 | Zbl 0219.60003

[12] G. B. Folland and E. Stein. Hardy Spaces on Homogeneous Groups. Princeton Univ. Press, Princeton, NJ, 1982. | MR 657581 | Zbl 0508.42025

[13] C. M. Goldie. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 (1991) 126-166. | MR 1097468 | Zbl 0724.60076

[14] A. K. Grincevičjus. Limit theorem for products of random linear transformations of the line. Litovsk. Mat. Sb. 15 (1975) 61-77, 241 (in Russian). | Zbl 0362.60018

[15] A. K. Grincevičjus. On a limit distribution for a random walk on lines. Litovsk. Mat. Sb. 15 (1975) 79-91, 243 (in Russian). | MR 448571 | Zbl 0373.60009

[16] H. Kesten. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973) 207-248. | MR 440724 | Zbl 0291.60029

[17] É. Le Page and M. Peigné. 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 1443957 | Zbl 0881.60018

[18] S. C. Port and C. J. Stone. Hitting time and hitting places for non-lattice recurrent random walks. J. Math. Mech. 17 (1967) 35-57. | MR 215375 | Zbl 0187.41202

[19] S. C. Port and C. J. Stone. Potential theory of random walks on Abelian groups. Acta Math. 122 (1969) 19-114. | MR 261706 | Zbl 0183.47201