Generalised regular variation of arbitrary order
Tom 90 / 2010
Streszczenie
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 $o(g(t))$ eventually leads to the relation $f(xt) - f(t) = h_1(x) g_1(t) + \cdots + h_n(x) g_n(t) + o(g_n(t))$, 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 $\boldsymbol{{g}} = (g_1, \ldots, g_n)'$. Under general assumptions, $\boldsymbol{{g}}$ itself must be regularly varying in the sense that $\boldsymbol{{g}}(xt) = x^{\boldsymbol{{B}}} \boldsymbol{{g}}(t) + o(g_n(t))$ for some upper triangular matrix $\boldsymbol{{B}} \in {\mathbb R}^{n \times n}$, and the vector of limit functions $\boldsymbol{{h}} = (h_1, \ldots, h_n)$ is of the form $\boldsymbol{{h}}(x) = \boldsymbol{{c}} \int_1^x u^{\boldsymbol{{B}}} u^{-1} {\rm d}u$ for some row vector $\boldsymbol{{c}} \in {\mathbb R}^{1 \times n}$. The uniform convergence theorem continues to hold. Based on this, representations of $f$ and $\boldsymbol{{g}}$ 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.
