Asymptotically balanced functions and the asymptotic behaviour of the complementary function and the Laplace transform (Q1122012)

Considering certain questions for sample extremes, \textit{L. de Haan} and \textit{S. Resnick} [Ann. Probab. 12, 588-608 (1984; Zbl 0538.60025)] called a measurable function \(f: {\mathbb{R}}^+\to {\mathbb{R}}\) asymptotically balanced if there exists an auxiliary function \(\alpha: {\mathbb{R}}^+\to {\mathbb{R}}^+\) such that: \[ \limsup_{t\to \infty}(f(tx)-f(x))/\alpha (t)<\infty \quad for\quad x>1, \] \[ \liminf_{t\to \infty}(f(tx)- f(x))/\alpha (t)>-\infty \quad for\quad x>0 \] and is eventually positive. The author discusses certain asymptotic relations between such a function f, its complementary function \(f^*(s)= \sup_{t>0}\{f(t)- ts\}\) and its log-Laplace transform \(\tilde f(s)=\log (s\int^{\infty}_{0}\exp (-st+f(t))dt).\) The results are technically too complicated to be stated here, but the following consequence of the main results gives some impression: Assume that for some nondecreasing function \(s: {\mathbb{R}}^+\to {\mathbb{R}}^+\) we have that: \(f(t)= \int^{t}_{0} s(x)dx+0(ts(t)),\quad t\to \infty,\) then \(\tilde f(u)= \int^{\infty}_{u} s^{\leftarrow}(x)dx+0(us^{\leftarrow}(u)),\) and vice versa, provided that s \((s^{\leftarrow}\) denotes the inverse) satisfies a variety of conditions, which are e.g. satisfied by \(s(t)=\log (t)/t.\)