Abstract: Consider the positive d-dimensional lattice Z₊^d (d≥2) with partial ordering ≤, let {X_K; 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 S_N =∑_K≤NX_K, N∈Z₊^d. Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums S_N 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).

Key words: The law of the logarithm, Random field, Hilbert space, Tail probability, Truncation method

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