Let Z₁, Z₂,⋯, be independent and identically distributed random variables and ᵢⱼₙ\ real numbers; put Tₙ = ∑ⁿi,j = 1 cᵢⱼₙZᵢZⱼ. This paper gives conditions under which the distribution of Tₙ - ETₙ converges to the distribution of ∑ Υₘ(Yₘ² - 1) with \Υₘ\ a real sequence and Y₁, Y₂,⋯ independent $N(0, 1)$ random variables. The results are applied to the calculation of the asymptotic distributions of test criteria of the form QₙW = ∑ F₀(Xₖₙ) - k/n + 1²W(k/n + 1) for testing the hypothesis that X₁ₙ, X₂ₙ,⋯, Xₙₙ are the order statistics of an independent sample from the distribution function F₀; here W is a weight function.
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Wet et al. (1973) studied this question.