Journal of Soviet Mathematics

, Volume 27, Issue 5, pp 2997–3002 | Cite as

Moderate deviations for densities in Rk

  • N. N. Amosova
  • Wolf -Dieter Richter


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


Moderate Deviation 
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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).Google Scholar
  2. 2.
    N. N. Amosova, “Probabilities of moderate deviations,” J. Sov. Math.,20, No. 3 (1982).Google Scholar
  3. 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

Authors and Affiliations

  • N. N. Amosova
  • Wolf -Dieter Richter

There are no affiliations available

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