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

TL;DR

This paper extends a key integral in distance correlation theory, providing a generalized formula involving truncated cosine expansions and establishing conditions for convergence.

Contribution

It introduces a generalized integral formula involving truncated cosine expansions, broadening the mathematical foundation of distance correlation analysis.

Findings

Derived a new integral formula valid under specific conditions.

Proved the independence of the constant from the truncation parameter.

Established convergence criteria for the generalized integral.

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 $m$.