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Pré-Publication, Document De Travail Année : 2019

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.) $d\times d$creal random matrices. Set $G_n = g_n g_{n-1} \ldots g_1$ and $X_n^x = G_n x/|G_n x|$, $n\geq 1$, where $|\cdot|$ is an arbitrary norm in $\mathbb R^d$ and $x\in\mathbb{R}^{d}$ is a starting point with $|x|=1$. For both invertible matrices and positive matrices, under suitable conditions we prove a Berry-Esseen type theorem and an Edgeworth expansion for the couple $(X_n^x, \log |G_n x|)$. 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 $X_n^x$. Cramér type moderate deviation expansions are derived for the couple $(X_n^x, \log |G_n x|)$ with a target function $\varphi$ on the Markov chain $X_n^x$. A local limit theorem with moderate deviations is also obtained.
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Dates et versions

hal-02173782 , version 1 (04-07-2019)

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  • HAL Id : hal-02173782 , version 1

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Hui Xiao, Ion Grama, Quansheng Liu. Berry-Esseen bound and precise moderate deviations for products of random matrices. 2019. ⟨hal-02173782⟩
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