On the limit distribution of the maximum of a random number of independent random variables

1964 ◽  
Vol 15 (3-4) ◽  
pp. 399-403 ◽  
Author(s):  
O. Barndorff-Nielsen
Keyword(s):  
Limit Distribution ◽  
Random Number ◽  
Random Variables ◽  
Independent Random Variables ◽  
Distribution Of The Maximum

scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (3) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

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

1973 ◽  
Vol 10 (01) ◽  
pp. 122-129 ◽  
Author(s):  
Janos Galambos
Keyword(s):  
Service Time ◽  
Joint Distribution ◽  
Random Number ◽  
Random Variables ◽  
Stochastic Independence ◽  
Fixed Integer ◽  
Number Of Components ◽  
Distribution Of The Maximum ◽  
Distribution Of Elements ◽  
Mild Assumption

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.

The Distribution of the Maximum of Partial Sums of Independent Random Variables

1950 ◽  
Vol 2 ◽  
pp. 375-384 ◽  
Author(s):  
Mark Kac ◽  
Harry Pollard
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Distribution Of The Maximum

1. The problem. It has been shown [1] that if Xi, + X2, … are independent random variables each of density then(1.1)

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