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August, 1973 Record Values and Maxima
Sidney I. Resnick
Ann. Probab. 1(4): 650-662 (August, 1973). DOI: 10.1214/aop/1176996892

Abstract

$\{X_n, n \geqq 1\}$ are $\operatorname{i.i.d.}$ random variables with continuous $\operatorname{df} F(x). X_j$ is a record value of this sequence if $X_j > \max \{X_1,\cdots, X_{j-1}\}$. We compare the behavior of the sequence of record values $\{X_{L_n}\}$ with that of the sample maxima $\{M_n\} = \{\max (X_1,\cdots, X_n)\}$. Conditions for the relative stability ($\operatorname{a.s.}$ and $\operatorname{i.p.}$) of $\{X_{L_n}\}$ are given and in each case these conditions imply the relative stability of $\{M_n\}$. In particular regular variation of $R(x) \equiv - \log (1 - F(x))$ is an easily verified condition which insures $\operatorname{a.s.}$ stability of $\{X_{L_n}\}, \{M_n\}$ and $\{\sum^n_{j=1} M_j\}$. Concerning limit laws, $X_{L_n}$ may converge in distribution without $\{M_n\}$ having a limit distribution and vice versa. Suitable differentiability conditions on $F(x)$ insure that both sequences have a limit distribution.

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Sidney I. Resnick. "Record Values and Maxima." Ann. Probab. 1 (4) 650 - 662, August, 1973. https://doi.org/10.1214/aop/1176996892

Information

Published: August, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0261.60024
MathSciNet: MR356186
Digital Object Identifier: 10.1214/aop/1176996892

Subjects:
Primary: 60F05

Keywords: Extreme values , limiting distributions , Maxima , Record values , regular variation , Relative stability

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 4 • August, 1973
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