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Generalised regular variation of arbitrary order

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Omey E., Segers J.

Banach Center Publications

2010

tom 90

nr 1

111-137

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.

Ograniczanie wyników

1 Banach Center Publications

1 Omey E.

1 Segers J.

1 2010

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Generalised regular variation of arbitrary order