The Annals of Probability

Weak Convergence to Extremal Processes

Sidney I. Resnick

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Abstract

$\{X_n, n\geqq 1\}$ are i.i.d. rv's with df $F$. Set $M_n = \max\{X_1, \cdots, X_n\}$. As a basic assumption, suppose normalizing constants $a_n > 0, b_n, n \geqq 1$ exist such that $\lim_{n\rightarrow\infty} P\lbrack M_n \leqq a_n x + b_n \rbrack = G(x)$, nondegenerate. Define the random function $Y_n(t) = (M_{\lbrack nt \rbrack} - b_n)/a_n$. By considering weak convergence of underlying two dimensional point processes, an alternate proof of the original Lamperti result that $Y_n \Rightarrow Y$ is given where $Y$ is an extremal-$G$ process. From the convergence of the point processes, other weak convergence results are shown. Let $x(t)$ be nondecreasing and $Nx(I)$ be the number of times $x$ jumps in time interval $I$. Then $Y_n^{-1} \Rightarrow Y^{-1}, NY_n \Rightarrow NY, NY_n^{-1} \Rightarrow NY^{-1}$. From these convergences emerge a variety of limit results for record values, record value times and inter-record times.

Article information

Source
Ann. Probab., Volume 3, Number 6 (1975), 951-960.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996221

Digital Object Identifier
doi:10.1214/aop/1176996221

Mathematical Reviews number (MathSciNet)
MR428396

Zentralblatt MATH identifier
0322.60024

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60J75: Jump processes 60J30 60B10: Convergence of probability measures

Keywords
Extreme values maxima record values weak convergence invariance principle Poisson process additive process extremal process

Citation

Resnick, Sidney I. Weak Convergence to Extremal Processes. Ann. Probab. 3 (1975), no. 6, 951--960. doi:10.1214/aop/1176996221. https://projecteuclid.org/euclid.aop/1176996221


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