Berry-Esseen bound and precise moderate deviations for products of random matrices
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866913697010548736 |
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| author | Xiao, Hui Grama, Ion Liu, Quansheng |
| author_facet | Xiao, Hui Grama, Ion Liu, Quansheng |
| contents | 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 $σ(G_n, x)= \log \frac{|G_n v|}{|v|}$, 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 $(X_n^x, σ(G_n, x))$. 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 as well as a local limit theorem with moderate deviations are proved for the couple $(X_n^x, σ(G_n, x))$ with a target function $φ$ on the Markov chain $X_n^x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_02438 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Berry-Esseen bound and precise moderate deviations for products of random matrices Xiao, Hui Grama, Ion Liu, Quansheng Probability 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 $σ(G_n, x)= \log \frac{|G_n v|}{|v|}$, 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 $(X_n^x, σ(G_n, x))$. 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 as well as a local limit theorem with moderate deviations are proved for the couple $(X_n^x, σ(G_n, x))$ with a target function $φ$ on the Markov chain $X_n^x$. |
| title | Berry-Esseen bound and precise moderate deviations for products of random matrices |
| topic | Probability |
| url | https://arxiv.org/abs/1907.02438 |