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Contents of PMS, Vol. 23, Fasc. 2,
pages 217 - 228
 

ALMOST SURE CONVERGENCE OF THE DISTRIBUTIONAL LIMIT THEOREM FOR ORDER STATISTICS

Liang Peng
Yongcheng Qi

Abstract: Let X ,n > 1,
 n be a sequence of independent and identically distributed random variables and X    < X   < ...< X
  n,1    n,2        n,n  denote the order statistics of X  ,...,X  .
  1      n For any sequence of integers (k )
  n with 1 < k < n
    n and lim     min(k ,n- k  +1)=   oo ,
  n-->o o      n     n if there exist constants a  > 0,b  (-  R
 n     n and some non-degenerate distribution function G such that (X    - b )/a
  n.kn    n  n  converges in distribution to G, then with probability one

       1   sum N 1 (Xn,k - bn    )
Nli-->mo o  log-N   n-I ----na---- < x  = G(x)  forallx  (-  C(G),
          n=1         n
where C(G) is the set of continuity points of G.

2000 AMS Mathematics Subject Classification: 60G15, 60F0S

Key words and phrases: Almost sure convergence, distributional limit theorem, order statistics.

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