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February, 1975 On Location and Scale Functions of a Class of Limiting Processes with Application to Extreme Value Theory
Ishay Weissman
Ann. Probab. 3(1): 178-181 (February, 1975). DOI: 10.1214/aop/1176996460

Abstract

Let $Y_n (n = 0, 1, \cdots)$ be a random variable and suppose that for suitably chosen constants $a_n$ and $b_n (b_n > 0)$ and each $t \in (0, \infty)$, the random variable $b_n Y_{\lbrack nt \rbrack} + a_n$ has a limit distribution function $G_t$. If $G_1$ is non-degenerate there are only two ways in which $G_t$ is related to $G_1$: there exist real constants $c$ and $\theta$ such that either $G_t(x) = G_1(c + t^\theta(x - c))$ for all $t > 0$ or else $G_t(x) = G_1(x + c \log t)$ for all $t > 0$. This result provides a very short derivation of the three types of extreme value limit distributions.

Citation

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Ishay Weissman. "On Location and Scale Functions of a Class of Limiting Processes with Application to Extreme Value Theory." Ann. Probab. 3 (1) 178 - 181, February, 1975. https://doi.org/10.1214/aop/1176996460

Information

Published: February, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0303.60021
MathSciNet: MR362458
Digital Object Identifier: 10.1214/aop/1176996460

Subjects:
Primary: 60G99
Secondary: 62E20 , 62G30

Keywords: extreme-value distribution functions , Limiting process , location and scale functions , type

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 1 • February, 1975
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