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Generalized regular variation of second order. (English) Zbl 0878.26002

Suppose \(f\) is a measurable real-valued function on \((0,\infty)\) bounded on \((0,k)\) for all \(k>0\). If there exists a positive function \(a(\cdot)\) such that \(\lim(t\to\infty)\) \((f(tx)- f(t))/a(t)\) exists for all \(x>0\), then the limit function \(\phi(x)\) is known to be of the form \(c_0\int^x_1 s^{\gamma-1}ds\) for some real \(c_0\), \(\gamma\). In the case that \(c_0\neq 0\) (and w.l.o.g. \(c_0=1\)), \(f\) is said to be of generalized regular variation. Motivated by considerations in probabilistic extreme value theory, the authors take this idea to “second-order”. That is, they assume that for a second positive “auxiliary” function \(a_1(t)\) there is a “non-trivial” limit \(\lim(t\to\infty)\) \((f(tx)- f(t)-a(t)\phi(x))/a_1(t)\), denoted by \(H(x)\).
The first problem is to find possible limit functions \(H\), and the choices of auxiliary functions \(a_1(\cdot)\). The subsequent theory is then developed in the pattern as for ordinary regularly varying functions.
Reviewer: E.Seneta (Sydney)

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
40E05 Tauberian theorems
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