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Limiting behavior of one sample rank order statistics with unbounded scores for nonstationary absolutely regular processes. (English) Zbl 0731.60006
Let \(X_{ni}\), \(1\leq i\leq n\), \(n\geq 1\), be real-valued r.v.’s with continuous distribution functions \(F_{ni}(x)\). The one-sample rank order statistic is given by \({\mathcal S}_{n,m}=\sum^{m}_{i=1}C_{ni} sgn(x) J(R_{n,m,i}/(m+1)),\) \(1\leq m\leq n\), where \(C_{ni}\) are constants defined through a continuous function h on (0,1) as \(C_{ni}=h(i/n)\), \(1\leq i\leq n\), \(n\geq 1\), J is a score function and \(R_{n,m,i}=\sum^{m}_{j=1}I[| X_{nj}| \leq | X_{ni}|],\) \(1\leq i\leq m\leq n\). Let \(Y_ n(s)={\mathcal S}_{n,[ns]}+(ns-[ns]){\mathcal S}_{n,[ns]+1},\) \(0\leq s\leq 1\). Assuming that the r.v.’s are absolutely regular with rate \(O(m^{-60(2- \delta)/\delta})\), \(\delta >0\), under certain conditions on h, J and \(F_{ni}\), the authors show that \(Y_ n(s)\), properly normalized, converges in the uniform topology over C[0,1] to \(\int^{s}_{0}h(u)dW(u)\), \(0\leq s\leq 1\), where W(t) is a Wiener process. The underlying assumptions allow \(C_{ni}\) and j(.) to be unbounded, thus extending an earlier result of the authors [J. Multivariate Anal. 30, No.2, 181-204 (1989; Zbl 0683.60007)].
MSC:
60B10 Convergence of probability measures
60F05 Central limit and other weak theorems
62G30 Order statistics; empirical distribution functions
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