On the asymptotic distribution of the sum of a random number of independent random variables

1957 ◽  
Vol 8 (1-2) ◽  
pp. 193-199 ◽  
Author(s):  
A. Rényi
Keyword(s):  
Asymptotic Distribution ◽  
Random Number ◽  
Random Variables ◽  
Independent Random Variables

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (01) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

scholarly journals On the probabilities of large deviations for sums of random number of independent random variables

1967 ◽  
Vol 7 (3) ◽  
pp. 513-516
Author(s):  
V. Statulevičius
Keyword(s):  
Large Deviations ◽  
Random Number ◽  
Random Variables ◽  
Independent 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: В. А. Статулявичюс. О вероятностях больших уклонений для суммы случайного числа независимых случайных величин V. Statulevičius. Didelių atsilenkimų tikimybių nepriklausomų atsitiktinių dydžių sumai su atsitiktiniu dėmenų skaičiumi klausimu

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