| Computational Mathematics & Applied Physics |
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| Precise asymptotics in the law of the logarithm for random fields in Hilbert space |
| FU Ke-ang, ZHANG Li-xin |
| Department of Mathematics, Zhejiang University, Hangzhou 310027, China |
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Abstract Consider the positive d-dimensional lattice Z+d (d≥2) with partial ordering ≤, let {XK; K∈Z+d be i.i.d. random variables taking values in a real separable Hilbert space (H, ||∙||) with mean zero and covariance operator ∑, and set partial sums SN =∑K≤NXK, N∈Z+d. Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums SN as ε↘0 by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of (Gut and Spǎtaru, 2003).
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Received: 31 July 2006
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