Large deviation expansions for the coefficients of random walks on the general linear group

2020/10/01 by Xiao, Hui, Grama, Ion, Liu, Quansheng
#37A30 #60B15 #60J05 #FOS: Mathematics #Primary 60F10 #Probability (math.PR) #Secondary 60B20

paper · doi:10.48550/arxiv.2010.00553

Abstract

Let (g_n)_{n≥ 1} be a sequence of independent and identically distributed elements of the general linear group GL(d, \mathbb R). Consider the random walk G_n: = g_n … g₁. Under suitable conditions, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients ⟨ f, G_n v ⟩, where f ∈ (\mathbb R^d)^* and v ∈ \mathbb R^d. In particular, our result implies the large deviation principle with an explicit rate function, thus improving significantly the large deviation bounds established earlier. Moreover, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients ⟨ f, G_n v ⟩ under the changed measure. Toward this end we prove the Hölder regularity of the stationary measure corresponding to the Markov chain G_n v /|G_n v| under the changed measure, which is of independent interest. In addition, we also prove local limit theorems with large deviations for the coefficients of G_n.

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