Functional limit theorems for linear statistics from sequential ranks

Abstract

Let X 1 ,..., X n be a sequence of continuously distributed independent random variables. The normalized ranks R kn and sequential ranks S k , k=1,...,n, are defined by $${\text{R}}_{{\text{kn}}} = \frac{1}{{\text{n}}}\sum\limits_{{\text{j}} = 1}^{\text{n}} {{\text{I}}\{ {\text{X}}_{\text{j}} < {\text{X}}_{\text{k}} \} ,} {\text{ S}}_{\text{k}} = \frac{1}{{\text{k}}}\sum\limits_{{\text{j = }}1}^{\text{n}} {{\text{I}}\{ {\text{X}}_{\text{j}} < {\text{X}}_{\text{k}} \} .} $$ The subject of the present paper is the asymptotic behavior, as n→∞, of the process $$\frac{1}{{\sqrt {\text{n}} }}\sum\limits_{{\text{k}} \leqq {\text{nt}}} {{\text{a}}({\text{S}}_{\text{k}} ),} {\text{ }}0 \leqq {\text{t}} \leqq 1,$$ for a∈L 2 (0, 1), $$\int\limits_0^1 {{\text{adn}} = 0} $$ . For suitable a, the limiting law of that process is expressed as solution of a stochastic equation under the hypothesis of identically distributed X 1,..., X n as well as under a class of contiguous alternatives, which contains the occurrence of a change point in the series of measurements.