# Generalised regular variation of arbitrary order

• Volume: 90, Issue: 1, page 111-137
• ISSN: 0137-6934

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## Abstract

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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 ≢ 0 and g > 0 such that f(xt) - f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) eventually leads to the relation f(xt) - f(t) = h₁(x)g₁(t) + ⋯ + hₙ(x)gₙ(t) + o(gₙ(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 = (g₁, ..., gₙ)’. Under general assumptions, g itself must be regularly varying in the sense that $g\left(xt\right)={x}^{B}g\left(t\right)+o\left(gₙ\left(t\right)\right)$ for some upper triangular matrix $B\in {ℝ}^{n×n}$, and the vector of limit functions h = (h₁, ..., hₙ) is of the form $h\left(x\right)=c{\int }_{1}^{x}{u}^{B}{u}^{-1}du$ for some row vector $c\in {ℝ}^{1×n}$. The uniform convergence theorem continues to hold. Based on this, representations of f and g can be derived in terms of simpler quantities. Moreover, the remainder terms in the asymptotic relations defining higher-order regular variation admit global, non-asymptotic upper bounds.

## How to cite

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Edward Omey, and Johan Segers. "Generalised regular variation of arbitrary order." Banach Center Publications 90.1 (2010): 111-137. <http://eudml.org/doc/282119>.

@article{EdwardOmey2010,
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 ≢ 0 and g > 0 such that f(xt) - f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) eventually leads to the relation f(xt) - f(t) = h₁(x)g₁(t) + ⋯ + hₙ(x)gₙ(t) + o(gₙ(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 = (g₁, ..., gₙ)’. Under general assumptions, g itself must be regularly varying in the sense that $g(xt) = x^\{B\}g(t) + o(gₙ(t))$ for some upper triangular matrix $B ∈ ℝ^\{n×n\}$, and the vector of limit functions h = (h₁, ..., hₙ) is of the form $h(x) = c∫_1^x u^\{B\} u^\{-1\}du$ for some row vector $c ∈ ℝ^\{1×n\}$. The uniform convergence theorem continues to hold. Based on this, representations of f and g can be derived in terms of simpler quantities. Moreover, the remainder terms in the asymptotic relations defining higher-order regular variation admit global, non-asymptotic upper bounds.},
author = {Edward Omey, Johan Segers},
journal = {Banach Center Publications},
keywords = {generalized regular variation; rate vector; characterization; uniform convergence; representation theorems; Potter bounds},
language = {eng},
number = {1},
pages = {111-137},
title = {Generalised regular variation of arbitrary order},
url = {http://eudml.org/doc/282119},
volume = {90},
year = {2010},
}

TY - JOUR
AU - Edward Omey
AU - Johan Segers
TI - Generalised regular variation of arbitrary order
JO - Banach Center Publications
PY - 2010
VL - 90
IS - 1
SP - 111
EP - 137
AB - 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 ≢ 0 and g > 0 such that f(xt) - f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) eventually leads to the relation f(xt) - f(t) = h₁(x)g₁(t) + ⋯ + hₙ(x)gₙ(t) + o(gₙ(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 = (g₁, ..., gₙ)’. Under general assumptions, g itself must be regularly varying in the sense that $g(xt) = x^{B}g(t) + o(gₙ(t))$ for some upper triangular matrix $B ∈ ℝ^{n×n}$, and the vector of limit functions h = (h₁, ..., hₙ) is of the form $h(x) = c∫_1^x u^{B} u^{-1}du$ for some row vector $c ∈ ℝ^{1×n}$. The uniform convergence theorem continues to hold. Based on this, representations of f and g can be derived in terms of simpler quantities. Moreover, the remainder terms in the asymptotic relations defining higher-order regular variation admit global, non-asymptotic upper bounds.
LA - eng
KW - generalized regular variation; rate vector; characterization; uniform convergence; representation theorems; Potter bounds
UR - http://eudml.org/doc/282119
ER -

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