Edgeworth expansion and large deviations for the coefficients of products of positive random matrices
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915159369318400 |
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| author | Xiao, Hui Grama, Ion Liu, Quansheng |
| author_facet | Xiao, Hui Grama, Ion Liu, Quansheng |
| contents | Consider the matrix products $G_n: = g_n \ldots g_1$, where $(g_{n})_{n\geq 1}$ is a sequence of independent and identically distributed positive random $d\times d$ matrices. Under the optimal third moment condition, we first establish a Berry-Esseen theorem and an Edgeworth expansion for the $(i,j)$-th entry $G_n^{i,j}$ of the matrix $G_n$, where $1 \leq i, j \leq d$. Using the Edgeworth expansion for $G_n^{i,j}$ under the changed probability measure, we then prove precise upper and lower large deviation asymptotics for the entries $G_n^{i,j}$ subject to an exponential moment assumption. As applications, we deduce local limit theorems with large deviations for $G_n^{i,j}$ and upper and lower large deviations bounds for the spectral radius $ρ(G_n)$ of $G_n$. A byproduct of our approach is the local limit theorem for $G_n^{i,j}$ under the optimal second moment condition. In the proofs we develop a spectral gap theory for the norm cocycle and for the coefficients, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_03158 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Edgeworth expansion and large deviations for the coefficients of products of positive random matrices Xiao, Hui Grama, Ion Liu, Quansheng Probability Consider the matrix products $G_n: = g_n \ldots g_1$, where $(g_{n})_{n\geq 1}$ is a sequence of independent and identically distributed positive random $d\times d$ matrices. Under the optimal third moment condition, we first establish a Berry-Esseen theorem and an Edgeworth expansion for the $(i,j)$-th entry $G_n^{i,j}$ of the matrix $G_n$, where $1 \leq i, j \leq d$. Using the Edgeworth expansion for $G_n^{i,j}$ under the changed probability measure, we then prove precise upper and lower large deviation asymptotics for the entries $G_n^{i,j}$ subject to an exponential moment assumption. As applications, we deduce local limit theorems with large deviations for $G_n^{i,j}$ and upper and lower large deviations bounds for the spectral radius $ρ(G_n)$ of $G_n$. A byproduct of our approach is the local limit theorem for $G_n^{i,j}$ under the optimal second moment condition. In the proofs we develop a spectral gap theory for the norm cocycle and for the coefficients, which is of independent interest. |
| title | Edgeworth expansion and large deviations for the coefficients of products of positive random matrices |
| topic | Probability |
| url | https://arxiv.org/abs/2209.03158 |