Extreme value theory for random walks on homogeneous spaces

Abstract

In this paper we study extreme value distributions for one parameter actions on homogeneous spaces of Lie groups. We study both shortest vectors in unimodular lattices, maximal distance excursions and closest distance returns of a one-parameter action. For certain sparse subsequences of the one-parameter action and by taking the maximum over a moving interval of indices we prove non-trivial estimates for the limiting distribution in all cases.

Spectral gap properties and limit theorems for some random walks and dynamical systems

“…For affine random walks it can be shown (see [31]) that Fréchet's law is still valid but with a parameter 0 < p < c, an inequality due to the clustering of extreme values, which corresponds to the mixing with speed properties of the weakly dependant sequence x n (see Lemma 4.7 below). For analogous results on different homogeneous spaces see [39].…”

Extreme value distributions for one-parameter actions on homogeneous spaces

“…Pollicott also showed how a logarithm law follows as a corollary of the above theorem, justifying the notion that EVL's are generalizations of logarithm laws. Indeed, as we demonstrated in [18], even an upper bound on the lim sup for the distribution of maxima along a sparse subsequence is sufficient to recover a logarithm law.…”

“…The proof of theorem 1.4 is similar to that of theorem 1.3 but notationally more complicated. For this reason we do not include the proof of theorem 1.4 in this paper but refer the reader to [19] for details.…”

On spectral gap properties and extreme value theory for multivariate affine stochastic recursions

“…Here, in this vein, we have the following logarithm law. If x is random, we observe that a logarithm law and a modified Fréchet law have been obtained in [21] for random walks on some homogeneous spaces of arithmetic character, using L 2 -spectral gap methods. Given a Borel subset A of U ′ 1 and a real number t > 1, we can also consider the hitting time τ x tA of the dilated set tA under by the process X x n (see [35] p. 290).…”

“…Then, using Theorem 3.1 and Proposition 3.4, it is shown below that the system (V Z + , τ, P ρ ) satisfies a multiple mixing condition with respect to Lipschitz functions. For a study of extreme value properties for random walks on some classes of homogeneous spaces, using L 2 -spectral gap methods, we refer to [21]. Since, using Proposition 2.4, the stationary process (X n ) n∈N satisfies also anticlustering, we see below that extreme value theory can be developed for (X n ) n∈N following the arguments of ([2] , [3]) which were developed under mixing conditions involving continuous functions.…”