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Statistics of the form Hₙ = d₈₉₍hₙ (Xᵢ, Xⱼ) are considered, where X₁, X₂, , are independent and identically distributed random variables, the diagonal terms, d₈₈₍, are equal to zero, and hₙ (x, y) is a symmetric real valued function. The asymptotic normality of such statistics is proven and the result then combined with work of Jogdeo on statistics that are weighted sums of bivariate functions of ranks to find sufficient conditions for asymptotic normality of permutation statistics derived from Hₙ.
Shapiro et al. (Sun,) studied this question.
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