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

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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 Székely, Rizzo and Bakirov (2007) to measure dependence between random variables. Let m be a positive integer and let ₘ(u), u ∈ R, be the truncated Maclaurin expansion of (u), where the expansion is truncated at the mth summand. For t, x ∈ Rᵈ, let t,x and x denote the standard Euclidean inner product and norm, respectively. We establish the integral formula: For α∈ C and x ∈ Rᵈ, ∫ᵣᵈ [ₘ(t,x) - (t,x)] dt/tᵈ⁺α = C(d,α) xα, with absolute convergence if and only if 2(m-1) < (α) < 2m. Moreover, the constant C(d,α) does not depend on m.