Extreme values of phase-type and mixed random variables with parallel-processing examples
| Autor: | Richard F. Serfozo, Sungyeol Kang |
|---|---|
| Rok vydání: | 1999 |
| Předmět: |
Statistics and Probability
Independent and identically distributed random variables General Mathematics 010102 general mathematics 01 natural sciences Stability (probability) Combinatorics 010104 statistics & probability Heavy-tailed distribution Sum of normally distributed random variables Generalized extreme value distribution Applied mathematics Mixture distribution 0101 mathematics Statistics Probability and Uncertainty Inverse distribution Mathematics Central limit theorem |
| Zdroj: | Journal of Applied Probability. 36:194-210 |
| ISSN: | 1475-6072 0021-9002 |
| DOI: | 10.1239/jap/1032374241 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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