A Generalization of an Integral Arising in the Theory of Distance Correlation

Abstract

We generalize an integral which arises in several areas in probability and statistics and which is at the core of the field of distance correlation, a concept developed by Sz\'ekely, Rizzo and Bakirov (2007) to measure dependence between random variables. Let $m$ be a positive integer and let ${\cos_m}(u)$, $u \in \mathbb{R}$, be the truncated Maclaurin expansion of ${\cos}(u)$, where the expansion is truncated at the $m$th summand. For $t, x \in \mathbb{R}^d$, let $\langle t,x\rangle$ and $\|x\|$ denote the standard Euclidean inner product and norm, respectively. We establish the integral formula: For $\alpha \in \mathbb{C}$ and $x \in \mathbb{R}^d$, $\int_{{\mathbb{R}}^d} [\cos_m(\langle t,x\rangle) - \cos(\langle t,x\rangle)] \,{\rm d}t/{\|t\|^{d+\alpha}} = C(d,\alpha) \, \|x\|^{\alpha}$, with absolute convergence if and only if $2(m-1) < \Re(\alpha) < 2m$. Moreover, the constant $C(d,\alpha)$ does not depend on $m$.