Consider $(g_n)_{n\geq 1}$ a sequence 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 $\langle f, G_n v \rangle$ 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. A local limit theorem with large deviations for the coefficients and large deviation expansions under the change of probability measure are also proved. 

　　  

　　Publication: 

　　The Annals of Probability 2023, Vol. 51, No. 4, 1380–1420 

　　http://dx.doi.org/10.1214/23-AOP1621 

　　  

　　Author: 

　　Hui Xiao 

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China 

　　Email: xiaohui@amss.ac.cn 

　　  

　　Ion Grama 

　　Univ Bretagne Sud, CNRS UMR 6205, LMBA 

　　Email: ion.grama@univ-ubs.fr 

　　  

　　Quansheng Liu 

　　Univ Bretagne Sud, CNRS UMR 6205, LMBA 

　　Email: quansheng.liu@univ-ubs.fr