The Annals of Probability

Regular Variation and the Stability of Maxima

R. J. Tomkins

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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

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

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


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

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


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

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