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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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##### References:
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