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Berry-Esseen type bounds for the left random walk on \({\mathrm{GL}_d}(\mathbb{R})\) under polynomial moment conditions. (English) Zbl 1519.60031

Let \(\varepsilon_1,\varepsilon_2,\ldots\) be an IID sequence drawn from \(\text{GL}_d(\mathbb{R})\), the group of invertible \(d\)-dimensional real matrices for some \(d\geq2\), with common distribution \(\mu\) assumed to be strongly irreducible and proximal. Letting \(\lVert\cdot\rVert\) denote the Euclidean norm on \(\mathbb{R}^d\) and \(\lVert A\rVert=\sup_{\lVert x\rVert=1}\lVert Ax\rVert\) for \(A\in\text{GL}_d(\mathbb{R})\), and with \(A_n=\varepsilon_n\cdots\varepsilon_1\), the authors establish Berry–Esseen inequalities for \(\log(\lVert A_n\rVert)\) under some moment assumptions on \(\mu\). That is, they give upper bounds on \[ \sup_{y\in\mathbb{R}}\left|\mathbb{P}(\log(\lVert A_n\rVert)-n\lambda_\mu\leq y\sqrt{n})-\Phi(y/s)\right| \] and \[ \sup_{\lVert x\Vert=1}\sup_{y\in\mathbb{R}}\left|\mathbb{P}(\log(\lVert A_nx\rVert)-n\lambda_\mu\leq y\sqrt{n})-\Phi(y/s)\right|\,, \] where \(\Phi\) is the distribution function of a standard Gaussian random variable, and \(\lambda_\mu\) and \(s\) are appropriate normalizing terms drawn from laws of large numbers-type results known in this setting. In particular, if \(\mu\) has finite \(q\)th moment for \(q\in(2,3]\) then upper bounds are of order \(\left(\frac{\log(n)}{n}\right)^{q/2-1}\). If \(\mu\) has finite fourth moment then the upper bounds are of order \(1/\sqrt{n}\).

MSC:

60F05 Central limit and other weak theorems
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks

References:

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