A characterization of multivariate regular variation.
The authors investigate the situations when the following two definitions of regular variation of a random vector are equivalent. First definition. The \(d\)-dimensional random vector \(X = (X_1,\dots,X_d)\) is regularly varying with index \(\alpha\geq 0\) if there exists a random vector \(Y\) with values in the unit sphere in \(R^d\) such that for all \(t > 0\) \[ \lim_{u\to\infty} \frac{P\{|X|>tu, X/|X|\in\cdot\}}{P\{|X|>u\}}= t^{-\alpha} P\{Y\in \cdot\}\;\text{ vague convergence,}\tag{1} \] where \(|\cdot|\) denotes a norm in \(R^d\). Second definition. The \(d\)-dimensional random vector \(X = (X_1,\dots,X_d)\) is regularly varying with index \(\alpha > 0\) if there exists a slowly varying function \(L\) such that for all \(x\in R^d\) \[ \lim_{u\to\infty}\;\frac{P\{(x,X)>u\}}{u^{-\alpha}L(u)} = w(x)\tag{2} \] and there exists a \(y\neq 0\) with \(w(y) > 0\), where \((\cdot,\cdot)\) denotes the usual inner product in \(R^d\).
scientific article; zbMATH DE number 1906175