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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)].
60F15 Strong limit theorems
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