Generalised regular variation of arbitrary order

Abstract

Let $f$ be a measurable, real function defined in a neighbourhood of infinity. The function $f$ is said to be of generalised regular variation if there exist functions $h \not\equiv 0$ and $g > 0$ such that $f(xt) - f(t) = h(x) g(t) + o(g(t))$ as $t \to \infty$ for all $x \in (0, \infty)$. Zooming in on the remainder term $o(g(t))$ leads eventually to a relation of the form $f(xt) - f(t) = h_1(x) g_1(t) + ... + h_n(x) g_n(t) + o(g_n(t))$, each $g_i$ being of smaller order than its predecessor $g_{i-1}$. The function $f$ is said to be generalised regularly varying of order $n$ with rate vector $\g = (g_1, >..., g_n)'$. Under general assumptions, $\g$ itself must be regularly varying in the sense that $\g(xt) = x^{\B} \g(t) + o(g_n(t))$ for some upper triangular matrix $\B \in \RR^{n \times n}$, and the vector of limit functions $\h = (h_1, >..., h_n)$ is of the form $\h(x) = \c \int_1^x u^\B u^{-1} \du$ for some row vector $\c \in \RR^{1 \times n}$. The usual results in the theory of regular variation such as uniform convergence and Potter bounds continue to hold. An interesting special case arises when all the rate functions $g_i$ are slowly varying, yielding $\Pi$-variation of order $n$, the canonical case being that $\B$ is equivalent to a single Jordan block with zero diagonal. The theory is applied to a long list of special functions.