Large deviation expansions for the coefficients of random walks on the general linear group
数学专题报告
报告题目(Title)：Large deviation expansions for the coefficients of random walks on the general linear group
报告人(Speaker)：肖惠 博士(中科院应用数学所)
地点(Place)：后主楼1124
时间(Time)：2023年4月12日(周三), 10:00-11:00
邀请人(Inviter)：何辉
报告摘要
Consider a sequence $(g_n)_{n\geq 1}$ of independent and identically distributed random matrices and the left random walk $G_n : = g_n \ldots g_1$ on the general linear group $GL(d, \mathbb R)$. Under suitable conditions, we establish Bahadur-Rao-Petrov type large deviation expansions for the coefficients of the product $G_n$, where $v \in \mathbb R^d$ and $f \in (\mathbb R^d)^*$. In particular, we obtain an explicit rate function in the large deviation principle, thus improving significantly the known large deviation bounds. Moreover, we prove local limit theorems with large deviations for the coefficients, and large deviation expansions under Cram\'er's change of probability measure. For the proofs we establish the H\"older regularity of the invariant measure of the Markov chain $(\mathbb R G_n v)$ under the changed probability, which is of independent interest. Joint work with I. Grama and Q. Liu.
