An inequality for a multidimensional characteristic function
Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 1, pp. 175-177
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Let $\xi $ be a vector-valued random variable in $\mathbf{R}^s$ and a corresponding density function $p_\xi(x)$ be “close” to the “standard”normal density. Under this condition an inequality for a characteristic function is proved. The inequality obtained is of interest for the problem of a lower estimator of the rate of convergence in the local limit theorem for densities. An analogous inequality for a lattice distribution was investigated in [N. G. Gamkrelidze, Litovsk. Mat. Sb., 7 (1967), pp. 405–408 (in Russian)] and was given in [V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag, Berlin, New York, 1975] and [Yu. V. Prohorov and Yu. A. Rozanov, Probability Theory: Basic Concepts, Limit Theorems, and Random Processes, Springer-Verlag, Berlin, New York, 1969].
Keywords:
vector-valued random variable, density function, standard normal density, characteristic function, local limit theorem.
@article{TVP_2000_45_1_a9,
author = {N. G. Gamkrelidze},
title = {An inequality for a multidimensional characteristic function},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {175--177},
publisher = {mathdoc},
volume = {45},
number = {1},
year = {2000},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2000_45_1_a9/}
}
N. G. Gamkrelidze. An inequality for a multidimensional characteristic function. Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 1, pp. 175-177. http://geodesic.mathdoc.fr/item/TVP_2000_45_1_a9/