On Small Deviations of Series of Weighted Random Variables

Боровков, А. А.; Ruzankin, Pavel

doi:10.1007/s10959-007-0130-x

Cited by 26 publications

(24 citation statements)

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“…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]). …”

Section: Introduction Letmentioning

confidence: 99%

“…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.…”

Section: Introduction Letmentioning

confidence: 99%

“…Theorem 1 [3,Theorem 2.1]. Assume that the function λ(u) is regularly varying at ∞ with an index −1/γ < −1 (and, therefore, the function λ −1 (x) is regularly varying at zero with the index γ ∈ (0, 1)).…”

Section: Introduction Letmentioning

confidence: 99%

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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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“…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]). …”

Section: Introduction Letmentioning

confidence: 99%

Section: Introduction Letmentioning

confidence: 99%

See 1 more Smart Citation

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

Rozovsky¹

2014

Theory Probab. Appl.

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“…Отправной точкой для нашего исследования являются результаты из [1]- [3] и [5], [6]. В этих работах (см.…”

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“…Заметим, что соответствующий результат из [3], как нам кажется, содержит неточность (функция, определенная в [3, (7.4)], не обязательно медленно меняется), и теорема 2 представляет его исправленный вари-ант.…”

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