Generalised regular variation of arbitrary order

TL;DR

This paper develops a comprehensive theory of generalized regular variation of arbitrary order, characterizing functions through rate vectors and matrix exponents, extending classical regular variation results and applying to various special functions.

Contribution

It introduces a new framework for generalized regular variation of any order, involving rate vectors and matrix exponents, expanding the classical theory.

Findings

The rate vector must be regularly varying with a matrix exponent.

The limit functions have a specific integral form involving the matrix exponent.

Uniform convergence and Potter bounds extend to this generalized setting.

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.