Asymptotique des valeurs extrêmes pour les marches aléatoires affines

Guivarcʼh, Yves; Le Page, Émile

Comptes Rendus. Mathématique, Tome 351 (2013) no. 17-18, pp. 703-705.

Nous considérons lʼespace Euclidien Rd et une marche aléatoire affine Xn sur Rd, gouvernée par une probabilité λ portée par le groupe affine H=Aff(Rd). Nous supposons que le sous-groupe de H engendré par le support de λ est « grand » et que la convolution par λ sur Rd admet une unique probabilité stationnaire η dont le support est non borné. Nous montrons la convergence en loi de certains processus ponctuels associés aux valeurs extrêmes de Xn. Les paramètres des lois limites sʼexpriment à lʼaide dʼune mesure homogène Λ sur Rd∖{0}, qui décrit lʼallure à lʼinfini de η et qui dépend essentiellement de la projection de λ sur le groupe linéaire de Rd. En particulier, les valeurs extrêmes normalisées de |Xn| suivent une loi de Fréchet, qui dépend simplement de Λ.

We consider the Euclidean space Rd and an affine random walk Xn on Rd, governed by a probability λ supported on the affine group H=Aff(Rd). We assume that the subgroup of H generated by the support of λ is “large” and that convolution by λ on Rd has a unique stationary probability η such that its support is unbounded. We show the convergence in law of certain point processes associated with the extreme values of Xn. The parameters of the limit laws are expressed in terms of a homogeneous measure Λ on Rd∖{0}, which describes the shape at infinity of η, and which depends essentially on the projection of λ on the linear group of Rd. In particular, the normalized extreme values of |Xn| follow a Fréchet law depending on Λ in a simple way.

[1] Bartkiewicz, K.; Jakubowski, A.; Mikosch, T.; Winterberger, O. Stable limits for sums of dependent infinite variance random variables, Probab. Theory Relat. Fields, Volume 150 (2011), pp. 337-372

[2] Basrak, B.; Segers, J. Regularly varying time series, Stoch. Process. Appl., Volume 119 (2009), pp. 1055-1080

[3] Collet, P. Statistics of closest returns for some non uniformly hyperbolic systems, Ergod. Theory Dyn. Syst., Volume 21 (2001), pp. 401-420

[4] Fréchet, M. Sur la loi de probabilité de lʼécart maximum, Ann. Soc. Math. Pol., Volume 6 (1927), pp. 93-116

[5] Gao, Z.; Guivarcʼh, Y.; Le Page, E. Stable laws and spectral gap properties for affine random walks, Ann. Inst. Henri Poincaré (2013) (à paraître, AIHP 1210-013 R1 A0)

[6] Guivarcʼh, Y.; Le Page, E. On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks, Ergod. Theory Dyn. Syst., Volume 28 (2008), pp. 423-446

[7] Guivarcʼh, Y.; Le Page, E. Spectral gap properties and asymptotics of stationary measures for affine random walks, 2013 | arXiv

[8] Hirata, M. Poisson law for Axiom A diffeomorphisms, Ergod. Theory Dyn. Syst., Volume 13 (1993), pp. 533-556

[9] Jaerish, J.; Kesseböhmer, M.; Stratmann, B.O. Fréchet law and Erdös–Philipp law for maximal cuspidal windings, 2012 | arXiv

[10] Kesten, H. Random difference equations and renewal theory for products of random matrices, Acta Math., Volume 131 (1973), pp. 207-248

[11] Kirseböm, M.S. Extreme value theory for random walks on homogeneous spaces, 2013 | arXiv

[12] Kleinbock, D.Y.; Margulis, G.A. Logarithm laws for flows on homogeneous spaces, Invent. Math., Volume 138 (1999), pp. 451-494

[13] Klüppelberg, C.; Pergamentchikov, S. Extremal behaviour of models with multivariate random recurrence representation, Stoch. Process. Appl., Volume 117 (2007), pp. 432-456

[14] Leadbetter, M.R.; Lindgren, G.; Rootzen, H. Extremes and Related Properties of Random Sequences and Processes, Springer, Berlin, 1983

[15] Pollicott, M. Limiting distributions for geodesic excursions on the modular surface, Contemp. Math., Volume 484 (2008), pp. 177-185

[16] Sullivan, D. Disjoint spheres, approximation by imaginary quadratic numbers and the logarithm law for geodesics, Acta Math., Volume 149 (1982), pp. 215-237