Title：

Large Deviations for Products of Random Matrices

Speaker:

Prof. Quansheng Liu, University of Bretagne Sud

Time：

December 8, 2020, 15:00-17:00 Beijing time

Teams Meeting Website：wstat.cn/meeting

Abstract: For a sum $S_n$ of independent random variables Bahadur and Rao (1960) and Petrov (1965) established equivalents for large deviation probabilities $ P(S_n > n(q + l))$, where $q$ is fixed and $l$ is vanishing as $n \rightarrow \infty$. These milestone results have numerous applications in a variety of problems in pure and applied probability. We study both invertible matrices and positive matrices, and obtain Bahadur-Rao-Petrov type results for $P( \log |G_nx| > n(q+l))$, where $G_n := g_n \cdots g_1$ is the product of independent and identically distributed $d\times d$ real random matrices $g_i$, $ |G_n x|$ denotes the norm of $G_nx$, $x$ is a starting point on the unit sphere in $\mathbb{R}^d$. As applications we improve previous results on large deviation principles for the matrix norm of $G_n$, and obtain a precise local limit theorem with large deviations.