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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 275–288 (Mi tvp844)

On multidimensional analogues of the S. N. Bernstein inequalities

A. V. Prokhorov

M. V. Lomonosov Moscow State University

Abstract: In the paper the method proposed in [1] of proving of multidimensional variants of the S. N. Bernstein inequalities is generalized. Instead of the estimate
$$\mathbf P(|Y_n|\ge r)\le C\exp\{-\frac{r^2}{8e^2L^2}\} \quad C=1+\frac{e^{5/12}}{\pi\sqrt2}\cdot\frac\Lambda{L^2},$$
where $Y_n$'s are normalized sums of independent identically distributed vectors with the restrictions (0.1), a more precise one is established:
$$\mathbf P(|Y_n|\ge r)\le C\exp\{-\frac{r^2}{8e^2L^2} \frac{\ln\frac{2L^2}\Lambda}{\ln2}\},\quad C=1+\frac{e^{13/24}}{2\pi}\sum_{s=2}^\infty s^{s/2}(e\ln2)^{-s}\sqrt s$$

A case is considered when it is possible to release from the dependence of the estimate on the dimensionality. It is indicated when the results obtained may be extended to the case of unequally distributed terms. Examples of application of various forms of the inequalities are given.

Full text: PDF file (626 kB)

English version:
Theory of Probability and its Applications, 1967, 13:2, 268–280

Bibliographic databases:

Citation: A. V. Prokhorov, “On multidimensional analogues of the S. N. Bernstein inequalities”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 275–288; Theory Probab. Appl., 13:2 (1967), 268–280

Citation in format AMSBIB
\Bibitem{Pro68} \by A.~V.~Prokhorov \paper On multidimensional analogues of the S.\,N.~Bernstein inequalities \jour Teor. Veroyatnost. i Primenen. \yr 1968 \vol 13 \issue 2 \pages 275--288 \mathnet{http://mi.mathnet.ru/tvp844} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=231425} \zmath{https://zbmath.org/?q=an:0169.20904} \transl \jour Theory Probab. Appl. \yr 1967 \vol 13 \issue 2 \pages 268--280 \crossref{https://doi.org/10.1137/1113030}