Asymptotic distributions of linear combinations of intermediate order statistics (Q2563789)

Let \(X,X_1,X_2,\dots\) be a sequence of independent non-degenerate random variables with a common distribution function \(F(x)=P\{X\leq x\}\), \(x\in\mathbb{R}\), and for each integer \(n\geq 1\) let \(X_{1,n}\leq \cdots\leq X_{n,m}\) denote the order statistics based on the sample \(X_1,\dots,X_n\). Let \(k_n\) be a sequence of positive numbers such that \[ k_n\to\infty \text{ and }k_n/n\to 0\text{ as }n\to\infty. \] When \(k_n\) are integers, the problem of investigating the asymptotic properties of the single intermediate order statistic \(X_{n-k_n,n}\) has considerable literature. \textit{S. Csörgő} and \textit{D. M. Mason} [Ann. Probab. 22, No. 1, 145-159 (1994; Zbl 0793.60020)] investigated the asymptotic distributions of the intermediate sums \[ \sum^{ \lceil bk_n \rceil}_{i=\lceil ak_n\rceil +1}X_{n+1 -i,n},\quad 0<a<b, \] where \(\lceil x\rceil\) is the smallest integer not smaller than \(x\). We generalize these results for the linear combinations \[ I_n(a,b):=\sum^{\lceil bk_n \rceil}_{i=\lceil ak_n\rceil+1} d_{n+1-i,n} f(X_{n+1-i,n}),\quad 0<a<b, \] of intermediate order statistics, where \(f\) is a known Borel-measurable function and \(d_{i,n}\), \(1\leq i\leq n\), are known constants. In the next section we formulate necessary and sufficient conditions for the existence of normalizing and centering constants \(A_n>0\) and \(C_n\) such that the sequence \(\{I_n(a,b)-C_n\}/A_n\) converges in distribution along subsequences of the positive integers to non-degenerate limits. In Section 3 our main theorems completely describe the possible subsequential limiting distributions of the properly centered and normalized linear combinations of intermediate order statistics.