Limit theorems for products of random matrices: from law of large numbers to large deviations
数学学科创建110周年系列报告
报告题目(Title)：Limit theorems for products of random matrices: from law of large numbers to large deviations
报告人(Speaker)：Ion Grama（Univ. Bretagne Sud, France）
地点(Place)：后主楼1220
时间(Time)：2025年5月23日（周五）11:00-12:00
邀请人(Inviter)：高志强
报告摘要
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 $\sigma(G_n, x)= \log \frac{|G_n v|}{|v|}$, 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 $(X_n^x, \sigma(G_n, x))$. 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 $X_n^x$. Large and moderate deviation expansions as well as a local limit theorem with moderate deviations will be considered for the couple $(X_n^x, \sigma(G_n, x))$ with a target function $\varphi$ on the Markov chain $X_n^x$.
We shall discuss applications to branching random walks and branching processes.
主讲人简介
Ion Grama 教授是法国南布列塔尼大学的一级终身教授，曾任南布列塔尼大学统计实验室主任。主要从事鞅，马氏过程， 数理统计和分支过程极限理论的研究．他与合作者有多篇论文发表在《Ann. Statist.》， 《Ann. Probab.》，《Probab. Theory Relat. Fields》，《Stochastic Process. Appl.》，《Bernoulli》等概率与统计方向的顶级刊物上．
