JournalsjemsVol. 24, No. 8pp. 2691–2750

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

  • Hui Xiao

    Universität Hildesheim, Germany
  • Ion Grama

    Université de Bretagne-Sud, Vannes, France
  • Quansheng Liu

    Université de Bretagne-Sud, Vannes, France
Berry–Esseen bound and precise moderate deviations for products of random matrices cover
Download PDF

This article is published open access under our Subscribe to Open model.

Abstract

Let (gn)n1{(g_{n})_{n\geq 1}} be a sequence of independent and identically distributed (i.i.d.) d×d{d\times d} real random matrices. For n1{n\geq 1} set Gn=gng1{G_n = g_n \ldots g_1}. Given any starting point x=RvPd1{x=\mathbb R v\in\mathbb{P}^{d-1}}, consider the Markov chain Xnx=RGnv{X_n^x = \mathbb R G_n v } on the projective space Pd1{\mathbb P^{d-1}} and define the norm cocycle by σ(Gn,x)=log(Gnv/v){\sigma(G_n, x)= \log (|G_n v|/|v|)}, for an arbitrary norm {|\cdot|} on Rd\smash{\mathbb R^{d}}. Under suitable conditions we prove a Berry–Esseen-type theorem and an Edgeworth expansion for the couple (Xnx,σ(Gn,x)){(X_n^x, \sigma(G_n, x))}. These results are established using a brand new smoothing inequality on complex plane, the saddle point method and additional spectral gap properties of the transfer operator related to the Markov chain Xnx{X_n^x}. Cramér-type moderate deviation expansions as well as a local limit theorem with moderate deviations are proved for the couple (Xnx,σ(Gn,x)){(X_n^x, \sigma(G_n, x))} with a target function φ{\varphi} on the Markov chain Xnx{X_n^x}.

Cite this article

Hui Xiao, Ion Grama, Quansheng Liu, Berry–Esseen bound and precise moderate deviations for products of random matrices. J. Eur. Math. Soc. 24 (2022), no. 8, pp. 2691–2750

DOI 10.4171/JEMS/1142