Asymptotic expansion of densities of the maxima of sums of random variables. Large deviations

1979 ◽  
Vol 19 (4) ◽  
pp. 508-516 ◽  
Author(s):  
B. Kryžienė
Keyword(s):  
Asymptotic Expansion ◽  
Large Deviations ◽  
Random Variables ◽  
Sums Of Random Variables

scholarly journals Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

2007 ◽  
Vol 44 (3) ◽  
pp. 670-684 ◽  
Author(s):  
Ph. Barbe ◽  
W. P. McCormick ◽  
C. Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Weibull Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Subexponential Distribution ◽  
Independent Random Variables ◽  
Sums Of Random Variables

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

Precise large deviations for sums of random variables with consistently varying tails

2004 ◽  
Vol 41 (01) ◽  
pp. 93-107 ◽  
Author(s):  
Kai W. Ng ◽  
Qihe Tang ◽  
Jia-An Yan ◽  
Hailiang Yang
Keyword(s):  
Large Deviations ◽  
Risk Model ◽  
Random Variables ◽  
Tail Probability ◽  
Arrival Process ◽  
Partial Sums ◽  
Precise Large Deviations ◽  
Sums Of Random Variables ◽  
Doubly Stochastic ◽  
Common Distribution Function

Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ > 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N(t), where N(·) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

scholarly journals Asymptotic expansions for the probabilities of large deviations for sums of random variables related to a Markov chain

1970 ◽  
Vol 10 (2) ◽  
pp. 359-366
Author(s):  
L. Saulis ◽  
V. Statulevičius
Keyword(s):  
Markov Chain ◽  
Large Deviations ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Sums Of Random Variables ◽  
Probabilities Of Large Deviations

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Л. И. Саулис, В. А. Статулявичус. Асимптотическое разложение для вероятностей больших уклонений сумм случайных величин, связанных в цепь Маркова L. Saulis, V. Statulevičius. Atsitiktinių dydžių, surištų į Markovo grandinę, didžiųjų nukrypimų asimptotinis dėstinys

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