The Annals of Probability

Regular Variation and the Stability of Maxima

R. J. Tomkins

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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 $\varepsilon > 0$.

Article information

Source
Ann. Probab., Volume 14, Number 3 (1986), 984-995.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992452

Digital Object Identifier
doi:10.1214/aop/1176992452

Mathematical Reviews number (MathSciNet)
MR841598

Zentralblatt MATH identifier
0593.60042

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48] 26A48: Monotonic functions, generalizations

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

Citation

Tomkins, R. J. Regular Variation and the Stability of Maxima. Ann. Probab. 14 (1986), no. 3, 984--995. doi:10.1214/aop/1176992452. https://projecteuclid.org/euclid.aop/1176992452


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