Stable laws and spectral gap properties for affine random walks

Gao, Zhiqiang; Guivarc’h, Yves; Le Page, Émile

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 319-348.

Nous considérons une relation de récurrence affine multidimensionelle à coefficients aléatoires et nous supposons que l’opérateur de Markov $P$ associé a une unique probabilité stationnaire. Nous montrons la propriété de trou spectral pour les opérateurs de Fourier correspondants sur certains espaces de fonctions Holdériennes, et nous en déduisons la convergence vers des lois stables pour les sommes de Birkhoff le long des trajectoires. Les paramètres des lois stables obtenues s’expriment à l’aide de quantités dépendant essentiellement de la partie multiplicative de $P$. La preuve est basée sur les propriétés spectrales de l’opérateur de Markov associé et l’homogénéité à l’infini de la mesure stationnaire.

We consider a general multidimensional affine recursion with corresponding Markov operator $P$ and a unique $P$-stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of $P$. Spectral gap properties of $P$ and homogeneity at infinity of the $P$-stationary measure play an important role in the proofs.

Classification : 60B20, 60E07, 60F05
Mots clés : stable laws, spectral gap, affine recursions

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