An Elementary Proof of Asymptotic Normality of Linear Functions of Order Statistics

Let X₁, ⋯, Xₙ be independent random variables, identically distributed with continuous distribution function F. Let X₍₁₎, ⋯, X₍ₙ₎ denote the corresponding order statistics and Fₙ the empiric distribution function, which we assume to be right continuous. Jung [4] found the asymptotic mean and variance of linear functions of the form Tₙ = n⁻¹ ∑ⁿᵢ₌₁ J(i/n)X₍ᵢ₎ = ∫^∞-∞ xJ(Fₙ(x)) dFₙ(x) when the function J has four bounded derivatives. More recently, it has been shown ([1], [2]) that under suitable restrictions {equation*}{1.1}L\{n1/2 T_n - ∫^∞-∞ xJ(F(x)) dF(x)\} → N(0, σ^2){equation*} where {equation*}{1.2}σ^2 = 2 ∫∫s < t J(F(s))J(F(t))F(s) 1 - F(t) ds dt.{equation*} Here N(0, σ²) denotes the normal distribution with mean zero and variance σ², and (1.1) uses a standard notation for convergence in distribution. The purpose of this note is to give a self-contained proof of (1.1) under the assumptions that the Xᵢ have finite mean and that $J'$ exists and is continuous and of bounded variation except at finitely many jumps of J. This theorem can be subsumed in a corrected version of Govindarajulu [2], where appeal is made to the results of [3]. The present proof is both more elementary and shorter. Chernoff, Gastwirth and Johns [1] do not require boundedness of J, but invoke compensating assumptions on F. Their methods of proof are quite different from those of [2] and of the present paper.