On limiting distributions of order statistics with variable ranks from stationary sequences. (English) Zbl 0584.60032

Cheng, Shihong

Ann. Probab. 13, 1326-1340 (1985).

Let \(\xi_ 1^{(n)}\leq...\leq \xi_ n^{(n)}\) be the order statistics of the r.v.’s \(\xi_ 1,...,\xi_ n\). In the case when \(\xi_ 1,...,\xi_ n\) are i.i.d. r.v.’s N. V. Smirnov [see Tr. Mat. Inst. Steklov 25 (1949; Zbl 0041.453); English translation in Am. Math. Soc. Transl. 67 (1952)] studied conditions for the weak convergence of the normalized sequence \((\xi^{(n)}_{K_ n}-b_ n)/a_ n\), \(n\geq 1\), to a certain d.f. \((K_ n\) is the so called variable rank sequence, i.e. such that \(\min (K_ n,n-K_ n)\to \infty).\)
In this paper similar results are obtained in the case when \(\xi_ n\), \(n\geq 1\), form a general stationary sequence. Two special kinds of stationary sequences are discussed.

Reviewer: R.Mnatsakanov

MSC:

60F05	Central limit and other weak theorems
60G10	Stationary stochastic processes
60G15	Gaussian processes

Keywords:

order statistics; weak convergence; variable rank sequence; stationary sequence

Citations:

Zbl 0041.453