Asymptotic Expansions for Sums of Random Variables and of Reciprocals of These with a Non-Normal Stable Limit Law

1985 ◽  
Vol 122 (1) ◽  
pp. 91-98 ◽  
Author(s):  
G. Christoph
Keyword(s):  
Asymptotic Expansions ◽  
Random Variables ◽  
Stable Limit ◽  
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.

Series expansions for convolutions of Pareto distributions

2015 ◽  
Vol 32 (1) ◽  
Author(s):  
Quang Huy Nguyen ◽  
Christian Y. Robert
Keyword(s):  
Infinite Series ◽  
Asymptotic Expansions ◽  
Value At Risk ◽  
Risk Measures ◽  
Random Variables ◽  
Series Expansions ◽  
Sums Of Random Variables ◽  
Scale Parameters ◽  
Pareto Distributions ◽  
Higher Order Terms

AbstractAsymptotic expansions for the tails of sums of random variables with regularly varying tails are mainly derived in the case of identically distributed random variables or in the case of random variables with the same tail index. Moreover, the higher-order terms are often given under the condition of existence of a moment of the distribution. In this paper, we obtain infinite series expansions for convolutions of Pareto distributions with non-integer tail indices. The Pareto random variables may have different tail indices and different scale parameters. We naturally find the same constants for the first terms as given in the previous asymptotic expansions in the case of identically distributed random variables, but we are now able to give the next additional terms. Since our series expansion is not asymptotic, it may be also used to compute the values of quantiles of the distribution of the sum as well as other risk measures such as the Tail Value at Risk. Examples of values are provided for the sum of at least five Pareto random variables and are compared to those determined via previous asymptotic expansions or via simulations.

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

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

2007 ◽  
Vol 44 (03) ◽  
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.

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