Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T13:32:33.476Z Has data issue: false hasContentIssue false
  • English
  • Français

The distribution of the maximum of a random number of random variables with applications

Published online by Cambridge University Press:  14 July 2016

Janos Galambos
Janos Galambos*
Affiliation:
Temple University, Philadelphia
Get access Rights & Permissions [Opens in a new window]

Abstract

The asymptotic distribution of the maximum of a random number of random variables taken from the model below is shown to be the same as when their number is a fixed integer. Applications are indicated to determine the service time of a system of a large number of components, when the number of components to be serviced is not known in advance. A much slighter assumption is made than the stochastic independence of the periods of time needed for servicing the different components. In our model we assume that the random variables can be grouped into a number of subcollections with the following properties: (i) the random variables taken from different groups are asymptotically independent, (ii) the largest number of elements in a subgroup is of smaller order than the overall number of random variables. In addition, a very mild assumption is made for the joint distribution of elements from the same group.

Keywords

ASYMPTOTIC DISTRIBUTIONRANDOM NUMBER OF RANDOM VARIABLESSERVICE TIMESYSTEM OF A LARGE NUMBER OF COMPONENTSDEPENDENT RANDOM VARIABLES
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 122 - 129
Copyright
Copyright © Applied Probability Trust 1973 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Barndorff-Nielsen, O. (1964) On the limit distribution of the maximum of a random number of independent random variables. Acta Math. Acad. Sci. Hungar. 15, 399–403.Google Scholar
[2] Galambos, J. (1972) On the distribution of the maximum of random variables. Ann. Math. Statist. 43, 516–521.CrossRefGoogle Scholar
[3] Gnedenko, B. V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Ann. of Math. 44, 423–453.Google Scholar
[4] Mogyoròdi, J. (1967) On the limit distribution of the largest term in the order statistic of a sample of random size. (In Hungarian) Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 17, 75–83.Google Scholar
[5] Rényi, A. (1961) A general method to prove theorems of probability and some of its applications. (In Hungarian) Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 11, 79–105.Google Scholar
[6] Rényi, A. (1963) On stable sequences of events. Sankhya Ser. A 25, 293–302.Google Scholar