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Contents of PMS, Vol. 32, Fasc. 1,
pages 117 - 130
 

COMPARISON THEOREMS FOR SMALL DEVIATIONS OF WEIGHTED SERIES

Leonid V. Rozovsky

Abstract: weighted series and obtain more refined versions of the known comparison results. In particular, the following consequence is obtained immediately from Theorem 2.1 of the paper.

Let a positive random variable X belong to the domain of attraction of a stable law with an index greater than one and let its distribution function be regularly varying at zero with an exponent β > 0 . If (X ) n n≥1 are independent copies of X , and (a ) n and (b ) n are positive summable sequences such that ∑ |1- a ∕b | < ∞, n≥1 n n then as + r → 0

∑ ∏ ∑ P ( anXn < r) ~ ( bn∕an)βP ( bnXn < r). n≥1 n≥1 n≥1

2000 AMS Mathematics Subject Classification: Primary: 60G50; Secondary: 60F99.

Keywords and phrases: Series of weighted i.i.d. positive random variables, small deviations, comparison theorems.

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