Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Authors:

Institution:

(1) Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China;(2) School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310035, P. R. China

Abstract:

Let {X, X _n; n ≥ 1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with covariance operator Σ. Set S _n = X ₁ + X ₂ + ... + X _n, n ≥ 1. We prove that, for b > −1,

$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(logn)^b }} {{n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }}} E\{ \left\| {S_n } \right\| - \sigma \varepsilon \sqrt {nlogn} \} _ + = \frac{{\sigma ^{ - 2(b + 1)} }} {{^{(2b + 3)(b + 1)} }}E\left\| Y \right\|^{2b + 3}$

holds if EX = 0, and E‖X‖²(log ‖X‖)^3b∨(b+4) < ∞, where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Σ, and σ ² denotes the largest eigenvalue of Σ. Project supported by National Natural Science Foundation of China (No. 10771192; 70871103)

Keywords:

the law of the logarithm moment convergence tail probability strong approximation

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