Regular Variation and the Stability of Maxima

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 $\varepsilon > 0$.