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Article Dans Une Revue Journal of the European Mathematical Society Année : 2021

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

Résumé

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed (i.i.d.) $d\times d$ real random matrices. For $n\geq 1$ set $G_n = g_n \ldots g_1$. Given any starting point $x=\mathbb R v\in\mathbb{P}^{d-1}$, consider the Markov chain $X_n^x = \mathbb R G_n v $ on the projective space $\mathbb P^{d-1}$ and the norm cocycle $\sigma(G_n, x)= \log \frac{|G_n v|}{|v|}$, for an arbitrary norm $|\cdot|$ on $\mathbb R^{d}$. Under suitable conditions we prove a Berry-Esseen type theorem and an Edgeworth expansion for the couple $(X_n^x, \sigma(G_n, x))$. 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 $X_n^x$. Cramer type moderate deviation expansions as well as a local limit theorem with moderate deviations are proved for the couple $(X_n^x, \sigma(G_n, x))$ with a target function $\varphi$ on the Markov chain $X_n^x$.
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Dates et versions

hal-03431385 , version 1 (22-11-2021)

Identifiants

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Hui Xiao, Ion Grama, Quansheng Liu. Berry–Esseen bound and precise moderate deviations for products of random matrices. Journal of the European Mathematical Society, inPress, ⟨10.4171/JEMS/1142⟩. ⟨hal-03431385⟩
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