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July, 1986 Regular Variation and the Stability of Maxima
R. J. Tomkins
Ann. Probab. 14(3): 984-995 (July, 1986). DOI: 10.1214/aop/1176992452

Abstract

The notion of regular variation of functions is generalized by defining $l(t) = \lim \inf_{x\rightarrow\infty} L(tx)/L(x), t > 0$, for any positive nondecreasing function $L$. It is shown that $l$ must obey one of: (i) $l(t) = +\infty$ for every $t > 1$; (ii) $l(t) > 1$ for every $t > 1$ and $l(t) \downarrow 1$ as $t \downarrow 1R$; or (iii) $l(t) = 1$ for some $t > 1$. Each of these classes is characterized in terms of the convergence or divergence of the integral $I(r, \delta) = \int_1^{\infty}\exp \{rL(\delta x) - L(x)\} dL(x)$ for $r \geq 1$, $\delta < 1$. Let $X_1, X_2, \ldots$ be i.i.d. random variables with distribution function $F$. Define $\mu_n = F^{-1}(1 - n^{-1}), M_n = \max(X_1,\ldots, X_n)$, and $L(x) = -\log(1 - F(x)). \{M_n\}$ is almost surely stable $\operatorname{iff} M_n/\mu_n \rightarrow 1$ a.s., and this is known to be equivalent to the convergence of $I(1, \delta)$ for every $\delta < 1$. Necessary and sufficient conditions for $\sum^\infty_{n=1} n^\alpha P\lbrack|(M_n/\mu_n) - 1| > \varepsilon\rbrack < \infty$ are presented, where $\alpha \geq -1$. In particular, that series converges $\operatorname{iff} I(\alpha + 2,(1 + \varepsilon)^{-1}) < \infty$. Moreover, the series $\sum^\infty_{n=1} n^\alpha P\lbrack|(M_n/\mu_n) - 1| > \varepsilon\rbrack$ converges for all $\varepsilon > 0$ and some $\alpha > -1 \operatorname{iff}$ it converges for every $\alpha > -1$ and every $\varepsilon > 0$.

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R. J. Tomkins. "Regular Variation and the Stability of Maxima." Ann. Probab. 14 (3) 984 - 995, July, 1986. https://doi.org/10.1214/aop/1176992452

Information

Published: July, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0593.60042
MathSciNet: MR841598
Digital Object Identifier: 10.1214/aop/1176992452

Subjects:
Primary: 60F15
Secondary: 26A12 , 26A48

Keywords: a.s. stability of maxima , complete convergence , generalized regular variation , i.i.d. random variables , regularly varying functions , Relative stability , Sample maxima

Rights: Copyright © 1986 Institute of Mathematical Statistics

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Vol.14 • No. 3 • July, 1986
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