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Moderate deviations for densities in Rk

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Abstract

Let {Xi=(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 Rk We assume that X1 has density p(x) and we denote by

the density of the normal law in Rk with mean 0=(0, ..., 0) and covariance matrix E and by Pn(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

.

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Literature cited

  1. 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).

  2. 2.

    N. N. Amosova, “Probabilities of moderate deviations,” J. Sov. Math.,20, No. 3 (1982).

  3. 3.

    A. D. Slastnikov, “Probabilities of moderate deviations,” Dokl. Akad. Nauk SSSR,238, No. 4, 814–815 (1978).

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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 7–13, 1982.

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Amosova, N.N., Richter, W.-. Moderate deviations for densities in Rk . J Math Sci 27, 2997–3002 (1984). https://doi.org/10.1007/BF01843543

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Keywords

  • Moderate Deviation