The Annals of Probability

Functional Limit Theorems for Extreme Values of Arrays of Independent Random Variables

Richard Serfozo

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Abstract

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.

Article information

Source
Ann. Probab., Volume 10, Number 1 (1982), 172-177.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993920

Digital Object Identifier
doi:10.1214/aop/1176993920

Mathematical Reviews number (MathSciNet)
MR637383

Zentralblatt MATH identifier
0482.60033

JSTOR
links.jstor.org

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G55: Point processes

Keywords
Order statistics extreme values functional limit theorem point process Poisson process extremal process

Citation

Serfozo, Richard. Functional Limit Theorems for Extreme Values of Arrays of Independent Random Variables. Ann. Probab. 10 (1982), no. 1, 172--177. doi:10.1214/aop/1176993920. https://projecteuclid.org/euclid.aop/1176993920


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