Functional limit theorems for linear statistics from sequential ranks (Q1069552)

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 \[ R_{kn}=\frac{1}{n}\sum^{n}_{j=1}I\{X_ j<X_ k\},\quad S_ k=\frac{1}{k}\sum^{k}_{j=1}I\{X_ j<X_ k\}. \] The subject of the present paper is the asymptotic behavior, as \(n\to \infty\), of the process \[ n^{-1/2}\sum_{k\leq nt}a(S_ k),\quad 0\leq t\leq 1, \] for \(a\in L^ 2(0,1)\), \(\int^{1}_{0}\) a \(dn=0\). 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.