论文标题
Edgeworth的扩展和大偏差，用于正随机矩阵产品系数
Edgeworth expansion and large deviations for the coefficients of products of positive random matrices
论文作者
论文摘要
考虑矩阵产品$ g_n：= g_n \ ldots g_1 $，其中$（g_ {n}）_ {n \ geq 1} $是一系列独立且分布的正随机$ d \ times d $矩阵。在最佳的第三刻条件下，我们首先建立了$（i，j）$ - th entrix $ g_n $的$（i，j）$的Edgeworth扩展，其中$ 1 \ leq i，j \ leq d $。在更改的概率度量下，使用$ g_n^{i，j} $的Edgeworth扩展，然后我们证明了条目的精确上下大偏差渐近级别$ g_n^{i，j} $，约为按指数力矩假设。作为应用程序，我们推断出具有$ g_n^{i，j} $的较大偏差的本地限制定理，以及$ g_n $的频谱半径$ρ（g_n）$的上和下大偏差范围。我们方法的副产品是$ g_n^{i，j} $在最佳的第二钟条件下的本地限制定理。在证据中，我们为规范合生子和系数开发了光谱差距理论，这具有独立的关注。
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.
