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Large deviation expansions for the coefficients of random walks on the general linear group
- Publication Year :
- 2020
-
Abstract
- Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed elements of the general linear group $GL(d, \mathbb R)$. Consider the random walk $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients $\langle f, G_n v \rangle$, where $f \in (\mathbb R^d)^*$ and $v \in \mathbb R^d$. In particular, our result implies the large deviation principle with an explicit rate function, thus improving significantly the large deviation bounds established earlier. Moreover, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients $\langle f, G_n v \rangle$ under the changed measure. Toward this end we prove the H\"{o}lder regularity of the stationary measure corresponding to the Markov chain $G_n v /|G_n v|$ under the changed measure, which is of independent interest. In addition, we also prove local limit theorems with large deviations for the coefficients of $G_n$.
- Subjects :
- Mathematics - Probability
Primary 60F10, 60B15, 37A30, Secondary 60B20, 60J05
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2010.00553
- Document Type :
- Working Paper