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Small deviations of modified sums of independent random variables

Authors:

L V Rozovsky

Institution:

(1) St. Petersburg Chemical Pharmaceutical Academy, St. Petersburg, Russia

Abstract:

Let S_n = X₁ + · · · + X _n, n ≥ 1, and S ₀ = 0, where X ₁, X ₂, . . . are independent, identically distributed random variables such that the distribution of S _n/B _n converges weakly to a nondeoenerate distribution F _α as n → ∞ for some positive B _n. We study asymptotic behavior of sums of the form

$\sum\limits_{n \geqslant 1} {f_n \,\,\text{P(}\frac{\text{1}}{{B_n }}R_{n}^{*} \leqslant \frac{{r}}{{\phi _n }}\text{)}} , \quad r \nearrow \infty ,$

where

$R_n^* = \mathop {\mathop {\max }\limits_{0 \leqslant k \leqslant n} \left( {S_k + d\left( {{k \mathord{\left/ {\vphantom {k n}} \right. \kern-\nulldelimiterspace} n}} \right)S_n } \right) - \mathop {\min }\limits_{0 \leqslant k \leqslant n} \left( {S_k + d\left( {{k \mathord{\left/ {\vphantom {k n}} \right. \kern-\nulldelimiterspace} n}} \right)S_n } \right),}\limits_{}$

a function d(t) is continuous at 0,1] and has power decay at zero,

$f_n \geqslant 0, \quad \sum\limits_{n \geqslant 1} {f_n = \infty ,}\quad and \quad \phi _n \nearrow \infty .$

Bibliography: 13 titles. Translated from Zapiski Nauchnylch Serninarov POMI, Vol. 361, 2008, pp. 109–122.

Keywords:

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