CENTRAL LIMIT THEOREM AND PRECISE LARGE DEVIATIONS FOR BRANCHING RANDOM WALKS WITH PRODUCTS OF RANDOM MATRICES
Résumé
We consider a branching random walk where particles give birth to children as a Galton-Watson process, which moves in $R^d$ with positions determined by the action of independent and identically distributed random matrices on the position of the parent. We are interested in asymptotic properties of the counting measure $Z^x_n$ which counts the number of particles of generation $n$ situated in a given region, when the process starts with one initial particle located at $x$. We establish a central limit theorem and a large deviation asymptotic expansion of Bahadur-Rao type for $ Z^x_n$ with suitable norming. An integral version of the large deviation result is also established. One of the key points in the proofs is the study of the fundamental martingale related to the spectral gap theory for products of random matrices. As a by-product, we obtain a sufficient and necessary condition for the non-degeneracy of the limit of the fundamental martingale, which extends the Kesten-Stigum type theorem of Biggins.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
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