# Generalised regular variation of arbitrary order

Banach Center Publications (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topEdward 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.