Generalised regular variation of arbitrary order
(2010) Banach Center Publications — n° 90, p. 111-137 (2010)
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- 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 h̸≡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)=h 1(x)g1(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)=xBg(t)+o(g n(t)) for some upper triangular matrix B∈R n×n, and the vector of limit functions h=(h1,…,h n) is of the form h(x)=c∫x 1u B u −1du 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.
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Omey, E., & Segers, J. (2010). Generalised regular variation of arbitrary order. Banach Center Publications, 90, 111-137. https://doi.org/10.4064/bc90-0-8 (Original work published 2010)