Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces
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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, 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.
$$
\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(logn)^b }}
{{n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} E\{ \left\| {S_n } \right\| - \sigma \varepsilon \sqrt {nlogn} \} _ + = \frac{{\sigma ^{ - 2(b + 1)} }}
{{^{(2b + 3)(b + 1)} }}E\left\| Y \right\|^{2b + 3}
$$
Keywords
the law of the logarithm moment convergence tail probability strong approximationMR(2000) Subject Classification
60F15 60G50Preview
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