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February, 1982 Functional Limit Theorems for Extreme Values of Arrays of Independent Random Variables
Richard Serfozo
Ann. Probab. 10(1): 172-177 (February, 1982). DOI: 10.1214/aop/1176993920


For an array $\{X_{ni}\}$ of independent, uniformly null random variables, several necessary and sufficient conditions are given for the convergence in distribution of its extremal process $\mathbf{M}_n = (M^1_n, M^2_n, \cdots)$ as $n \rightarrow \infty$, where $M^k_n(t) = k$th largest $\{X_{ni}: i/n \leq t\}, t > 0$. It is shown that if $\mathbf{M}_n$ converges, then its limit is an extremal process of a Poisson process on the plane. The limit cannot be an extremal process of a non-Poisson, infinitely divisible point process, which is possible for certain stationary variables. A characterization of the convergence of $\mathbf{M}_n$, without the uniformly null assumption, is also given.


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Richard Serfozo. "Functional Limit Theorems for Extreme Values of Arrays of Independent Random Variables." Ann. Probab. 10 (1) 172 - 177, February, 1982.


Published: February, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0482.60033
MathSciNet: MR637383
Digital Object Identifier: 10.1214/aop/1176993920

Primary: 60F17
Secondary: 60G55

Keywords: Extremal process , Extreme values , Functional limit theorem , order statistics , point process , Poisson process

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 1 • February, 1982
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