## Abstract

Let {X

^{i}=(X_{1}^{i}, ..., X_{k}^{i}), i=1, 2, ...} be a sequence of independent identically distributed random vectors with values in the k-dimensional Euclidean space R^{k}We assume that X^{1}has density p(x) and we denote by the density of the normal law in R^{k}with mean 0=(0, ..., 0) and covariance matrix E and by P_{n}(x) the density of the random vector\(Z_n = \frac{1}{{\sqrt n }}\left( {X^1 + ... + X^n } \right)\). In the paper one finds conditions which are necessary and sufficient in order to have the relation\(P_n \left( x \right) = \varphi _{o, E} \left( x \right)\left( {1 + o\left( 1 \right)} \right),n \to \infty \) uniformly with respect to x, x ∈, where.## Keywords

Moderate Deviation
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## Literature cited

- 1.N. N. Amosova, “On the probabilities of moderate deviations for sums of independent random variables,” Teor. Veroyatn. Primen.,24, No. 4, 858–865 (1979).Google Scholar
- 2.N. N. Amosova, “Probabilities of moderate deviations,” J. Sov. Math.,20, No. 3 (1982).Google Scholar
- 3.A. D. Slastnikov, “Probabilities of moderate deviations,” Dokl. Akad. Nauk SSSR,238, No. 4, 814–815 (1978).Google Scholar

## Copyright information

© Plenum Publishing Corporation 1984