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Convergence rates for probabilities of moderate deviations for multidimensionally indexed random variables. (English) Zbl 1195.60045
Summary: Let $$\{X,X_{\overline n};\overline n\in\mathbb Z_+^d\}$$ be a sequence of i.i.d. real-valued random variables, and $$S_{\overline n}=\sum_{\overline k\leq\overline n}X_{\overline k}$$, $$\overline n\in\mathbb Z_+^d$$. Convergence rates of moderate deviations are derived; that is, the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of the series $$\sum_{\overline n} b(\overline n)\psi^2(a(\overline n))P\{|S_{\overline n}| \geq a(\overline n)\geq (a(\overline n))\}$$, where $$a(\overline n)=n_1^{1/\alpha_1}\cdots n_d^{1/\alpha_d}$$, $$b(\overline n)=n_1^{\beta_1}\cdots n_d^{\beta_d}$$, $$\phi$$ and $$\psi$$ are taken from a broad class of functions. These results generalize and improve some results of D. Li, X. Wang, and M. B. Rao [Int. J. Math. Math. Sci. 15, No. 3, 481–497 (1992; Zbl 0753.60028)] and some previous work of A. Gut [Ann. Probab. 8, 298–313 (1980; Zbl 0429.60022)].
##### MSC:
 60F15 Strong limit theorems
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##### References:
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