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 Teor. Veroyatnost. i Primenen., 1969, Volume 14, Issue 4, Pages 708–715 (Mi tvp1482)

Short Communications

On the distribution of the maximum of cumulative sums of independent random variables

V. B. Nevzorov, V. V. Petrov

Abstract: Let $X_1,…,X_n$ be independent random variables, $S_k=\sum_{j=1}^kX_j$, $\overline S_n=\max\limits_{1\le k\le n}S_k$. Set
$$G(x)= \begin{cases} \sqrt{\frac2\pi}\int_0^xe^{t^2/2} dt,&x>0, 0,&x\le0. \end{cases}$$
An estimate for $\sup|\mathbf P(\overline S_n<bx)-G(x)|$, where $b$ is an arbitrary positive number, is obtained without assumptions about the existence of moments. Some corrolaries are derived from this result. For example, if $\mathbf EX_k=0$ for all $k$ and $q_n^2=\sum_{k=1}^n\mathbf EX_k^2<\infty$, then
$$\sup_x|\mathbf P(\overline S_n<q_nx)-G(x)|<\frac{\Lambda_n(\varepsilon)}{\varepsilon^2}+12\varepsilon$$
for any $\varepsilon>0$. Here $\Lambda_n(\varepsilon)$ is the Lindeberg ratio defined by (10).

Full text: PDF file (360 kB)

English version:
Theory of Probability and its Applications, 1969, 14:4, 682–687

Bibliographic databases:

Citation: V. B. Nevzorov, V. V. Petrov, “On the distribution of the maximum of cumulative sums of independent random variables”, Teor. Veroyatnost. i Primenen., 14:4 (1969), 708–715; Theory Probab. Appl., 14:4 (1969), 682–687

Citation in format AMSBIB
\Bibitem{NevPet69} \by V.~B.~Nevzorov, V.~V.~Petrov \paper On the distribution of the maximum of cumulative sums of independent random variables \jour Teor. Veroyatnost. i Primenen. \yr 1969 \vol 14 \issue 4 \pages 708--715 \mathnet{http://mi.mathnet.ru/tvp1482} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=263139} \zmath{https://zbmath.org/?q=an:0204.51203|0188.23802} \transl \jour Theory Probab. Appl. \yr 1969 \vol 14 \issue 4 \pages 682--687 \crossref{https://doi.org/10.1137/1114083} 

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This publication is cited in the following articles:
1. Borodin A.N. Davydov Yu.A. Nevzorov V.B., “On the History of the St. Petersburg School of Probability and Statistics. III. Distributions of Functionals of Processes, Stochastic Geometry, and Extrema”, Vestn. St Petersb. Univ.-Math., 51:4 (2018), 343–359
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