Precise large deviation asymptotics for products of random matrices

Abstract

Let (gn) n 1 be a sequence of independent identically distributed d × d real random matrices with Lyapunov exponent γ. For any starting point x on the unit sphere in R d , we deal with the norm |Gnx|, where Gn := gn . . . g1. The goal of this paper is to establish precise asymptotics for large deviation probabilities P(log |Gnx| n(q +l)), where q > γ is fixed and l is vanishing as n → ∞. We study both invertible matrices and positive matrices and give analogous results for the couple (X x n , log |Gnx|) with target functions, where X x n = Gnx/|Gnx|. As applications we improve previous results on the large deviation principle for the matrix norm Gn and obtain a precise local limit theorem with large deviations.

Random walks on hyperbolic spaces: Concentration inequalities and probabilistic Tits alternative

“…Remark 1. 14 Similarly to Corollary 1.8, the estimate in Proposition 1.13 allows one to obtain a global lower bound (less explicit in its constants compared to the aforementioned corollary) for the rate function of log-norms of random matrix products studied in [55,60] (see also [55,Corollary 4.17]).…”

Berry–Esseen bounds and moderate deviations for random walks on GLd(R)

“…Moderate deviations have not yet been studied neither for G n nor for ρ(G n ), to the best of our knowledge. For large deviations, the upper tail large deviation principle for G n has been established by Sert [24] and [25] under different conditions; it is conjectured in [24] that the usual large deviation principle would hold for ρ(G n ).…”

“…From the large deviation bounds for log G k (see [5] or [25]), we have that for any q > λ, there exist constants c, C > 0 such that for any k 1,…”

Berry-Esseen bound and precise moderate deviations for products of random matrices

“…From now on, for any integrable function h : R → C, denote its Fourier transform by h(t) = R e −ity h(y)dy, t ∈ R. If h is integrable on R, then using the inverse Fourier transform gives h(y) = Following [37], we construct a density function ρ T which plays an important role in establishing a new smoothing inequality. Specifically, on the real line define…”

“…For products of random matrices, precise large deviations asymptotics have been considered e.g. by Le Page [29], Buraczewski and Mentemeier [6], Guivarc'h [16], Benoist and Quint [3], Sert [33], Xiao, Grama and Liu [37]. For moderate deviations, very little results are known.…”