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On the lower tail probabilities of some random series. / Lifshits, M. A.

In: Annals of Probability, Vol. 25, No. 1, 01.01.1997, p. 424-442.

Research output: Contribution to journal › Article › peer-review

Harvard

Lifshits, MA 1997, 'On the lower tail probabilities of some random series', Annals of Probability, vol. 25, no. 1, pp. 424-442. https://doi.org/10.1214/aop/1024404294

APA

Lifshits, M. A. (1997). On the lower tail probabilities of some random series. Annals of Probability, 25(1), 424-442. https://doi.org/10.1214/aop/1024404294

Vancouver

Lifshits MA. On the lower tail probabilities of some random series. Annals of Probability. 1997 Jan 1;25(1):424-442. https://doi.org/10.1214/aop/1024404294

Author

Lifshits, M. A. / On the lower tail probabilities of some random series. In: Annals of Probability. 1997 ; Vol. 25, No. 1. pp. 424-442.

BibTeX

@article{0180a6054d0b451b9566aa019fdc24ee,
title = "On the lower tail probabilities of some random series",
abstract = "The behavior of tail probabilities P{S ≤ r}, r → 0 is investigated, where S is a series S = ∑λjZj generated by some sequence of positive numbers {λj} and by a sequence {Zj} of independent copies of a positive random variable Z. We present the exact asymptotic expression for P{S ≤ r} by means of Laplace transform Λ(γ) = Eexp{-γS} under weak assumptions on the behavior of the tail probabilities of Z in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of S are investigated.",
keywords = "Central limit theorem, Laplace transform, Lower tail probabilities, Small balls, Sums of independent variables",
author = "Lifshits, {M. A.}",
year = "1997",
month = jan,
day = "1",
doi = "10.1214/aop/1024404294",
language = "English",
volume = "25",
pages = "424--442",
journal = "Annals of Probability",
issn = "0091-1798",
publisher = "Institute of Mathematical Statistics",
number = "1",

}

RIS

TY - JOUR

T1 - On the lower tail probabilities of some random series

AU - Lifshits, M. A.

PY - 1997/1/1

Y1 - 1997/1/1

N2 - The behavior of tail probabilities P{S ≤ r}, r → 0 is investigated, where S is a series S = ∑λjZj generated by some sequence of positive numbers {λj} and by a sequence {Zj} of independent copies of a positive random variable Z. We present the exact asymptotic expression for P{S ≤ r} by means of Laplace transform Λ(γ) = Eexp{-γS} under weak assumptions on the behavior of the tail probabilities of Z in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of S are investigated.

AB - The behavior of tail probabilities P{S ≤ r}, r → 0 is investigated, where S is a series S = ∑λjZj generated by some sequence of positive numbers {λj} and by a sequence {Zj} of independent copies of a positive random variable Z. We present the exact asymptotic expression for P{S ≤ r} by means of Laplace transform Λ(γ) = Eexp{-γS} under weak assumptions on the behavior of the tail probabilities of Z in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of S are investigated.

KW - Central limit theorem

KW - Laplace transform

KW - Lower tail probabilities

KW - Small balls

KW - Sums of independent variables

UR - http://www.scopus.com/inward/record.url?scp=0031519750&partnerID=8YFLogxK

U2 - 10.1214/aop/1024404294

DO - 10.1214/aop/1024404294

M3 - Article

AN - SCOPUS:0031519750

VL - 25

SP - 424

EP - 442

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1

ER -

ID: 43811445