论文标题
通用线性群体随机步行系数的大偏差扩展
Large deviation expansions for the coefficients of random walks on the general linear group
论文作者
论文摘要
令$（g_n）_ {n \ geq 1} $为一系列独立且相同分布的元素$ gl（d，\ mathbb r）$。考虑随机步行$ g_n：= g_n \ ldots g_1 $。在适当的条件下，我们为系数$ \ langle f，g_n v \ rangle $建立了bahadur-rao-petrov型大偏差扩展，其中$ f \ in（\ sathbb r^d）^*$ and $ v \ in \ mathbb r^d $。特别是，我们的结果意味着具有明确速率函数的较大偏差原理，从而显着改善了早期建立的较大偏差界限。此外，我们在更改的度量下建立了系数$ \ langle f，g_n v \ rangle $的Bahadur-rao-petrov型大偏差扩展。为此，我们证明了与马尔可夫链$ g_n v /| g_n v | $相对应的固定度量的Hölder规律性，这是独立的利益。此外，我们还证明了本地限制定理，其系数$ g_n $的偏差很大。
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 Hölder 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 $G_n$.
