Precise rates in the law of the logarithm in the Hilbert space
type: Item
instance of: scholarly article
title: Precise rates in the law of the logarithm in the Hilbert space
zbMATH Open document ID: 1109.60036
author: Wei Huang
author: Lixin Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-18T00:00:00Z
review text: Let \(\{X,X_n\); \(n\geq 1\}\) be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (\(\mathbb H,\|\cdot\|)\) with covariance operator \(\Sigma\), and set \(S_n=X_1+\cdots+X_n\), \(n\geq 1\). Let \(a_n=o(\sqrt{n/\log n})\). We prove that, for any \(1< r < 3/2\) and \(a>-d/2\), \[ \begin{multlined} \lim_{\varepsilon\searrow\sqrt{r-1})}[\varepsilon^2-(r-1)]^{a+d/2}\sum^\infty_{n=1}n^{r-2}(\log n)^a{\mathbf P}\{\|S_n\|\geq \sigma\phi(n)\varepsilon + a_n\}\\ =\Gamma^{-1}(d/2)K(\Sigma)(r-1)^{(d-2)/2}\Gamma(a+d/2)\end{multlined} \] holds if \[ {\mathbf E} X=0,\quad {\mathbf E}[\|X\|^{2r}(\log\| X\|)^{a-r}]<\infty. \]
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 2157713
zbMATH Keywords: Complete convergence
zbMATH Keywords: Tail probabilities of sums of i.i.d. random variables
zbMATH Keywords: Strong approximation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.09.052
OpenAlex ID: W2256112765
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DOI: 10.1016/J.JMAA.2004.09.052
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description: scientific article; zbMATH DE number 2157713