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Convergence rates for probabilities of moderate deviation for sums of random variables indexed by \(Z_ +^ d\). (English) Zbl 0822.60023
Let \(X\), \(X_{\overline{n}}\), \(\overline {n} \in \mathbb{Z}^ d_ +\), be a sequence of i.i.d. real random variables, and \(S_{\overline {n}} = \sum_{\overline {k} \leq \overline {n}} X_{\overline {k}}\), \(\overline{n} \in Z^ d_ +\). Convergence rates of moderate deviation are derived, i.e., the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain sufficient conditions and necessary conditions for the convergence of series \[ \sum_{\overline {n}} b(\overline{n}) P \{| S_{\overline {n}} | \geq \varepsilon(a^ 2 (\overline {n}) \log a^ 2 (\overline {n}))^{1/2}\}, \] where \(a(\overline {n}) = n^{1/\alpha_ 1}_ 1 \dots n^{1/\alpha_ d}_ d\), \(b(\overline {n}) = n^{\beta_ 1}_ 1 \dots n^{\beta_ d}_ d\). We also give the corresponding results for \(P\{ | S_{\overline{n}}| \geq \varepsilon(a^ 2 (\overline{n}) \log_ 2 a^ 2 (\overline {n}))^{1/2}\}\) which give connection to the law of the iterated logarithm. These results generalize and improve some previous work of A. Gut [Ann. Probab. 8, 298-313 (1980; Zbl 0429.60022)].
Reviewer: D.Deng (Changchun)

MSC:
60F10 Large deviations
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F99 Limit theorems in probability theory
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