On small deviations of series of weighted positive random variables

Rozovsky, L. V.

doi:10.1007/s10958-011-0413-8

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Cited by 8 publications

(3 citation statements)

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“…where {a j } ∈ ℓ 1 (R + ) and {ζ j } are non-negative i.i.d. random variables, have been studied in [1,7,22,27,28,18]. In particular, we present the following theorem (Theorem 3.1 of [7]) without proof.…”

Section: 1mentioning

confidence: 99%

Small deviations for time-changed Brownian motions and applications to second-order chaos

Dobbs¹,

Melcher

2014

Electron. J. Probab.

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Section: 1mentioning

confidence: 99%

Small deviations for time-changed Brownian motions and applications to second-order chaos

Dobbs¹,

Melcher

2014

Electron. J. Probab.

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Section: Similar Problems Were Studied By a Number Of Authors (An Extmentioning

confidence: 99%

“…There (see also [9]) an explicit form of asymptotics of − log P(S < r) for r → 0 was obtained under minimal a priori assumptions, provided that the sequence {λ j } satisfies certain additional conditions. We formulate the corresponding conclusions and do so in terms of Laplace transforms of X and S. In our opinion, this formulation is simpler and shorter.…”

Section: Introduction Letmentioning

confidence: 99%

Small Deviations of Series of Independent Nonnegative Random Variables with Smooth Weights

Rozovsky¹

2014

Theory Probab. Appl.

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We study the logarithmic small deviations of j 1 λ j X j , where {X j } are independent identically distributed nonnegative random variables and {λ j } is a sequence of positive and nonincreasing numbers satisfying certain conditions of regularity. Introduction. Letwhere {λ j } is a sequence of nonincreasing positive numbers and {X i } are independent copies of a nonnegative random variable X with a distribution function V (x), and P(S < ∞) = 1. We examine the behavior of − log P(S < r) for r → 0. Similar problems were studied by a number of authors (an extensive list of references can be found in [1], [2], [3], [4]).In the present paper we take the results from [1], [2], [3] and [5], [6] as a base. There (see also [9]) an explicit form of asymptotics of − log P(S < r) for r → 0 was obtained under minimal a priori assumptions, provided that the sequence {λ j } satisfies certain additional conditions.We formulate the corresponding conclusions and do so in terms of Laplace transforms of X and S. In our opinion, this formulation is simpler and shorter. The equivalent recalculation in terms of distribution functions is not a problem (see section 6).Set

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Probabilities of Small Deviations of the Weighted Sum of Independent Random Variables with Common Distribution That Decreases at Zero Not Faster Than a Power

Rozovsky

2016

J Math Sci

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On small deviations of series of weighted positive random variables

Cited by 8 publications

References 11 publications

Small deviations for time-changed Brownian motions and applications to second-order chaos

Small deviations for time-changed Brownian motions and applications to second-order chaos

Small Deviations of Series of Independent Nonnegative Random Variables with Smooth Weights

Probabilities of Small Deviations of the Weighted Sum of Independent Random Variables with Common Distribution That Decreases at Zero Not Faster Than a Power

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