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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 23, Fasc. 1,
pages 153 - 172
 

ON STABILITY OF TRIMMED SUMS

Tien-Chung Hu
Chiung-Yu Huang
Andrew Rosalsky

Abstract: Let (X ,n > 1)
  n be a sequence of i.i.d. random variables and let (a ,n > 1)
  n and (bn,n > 1) be sequences of constants where 0 < bn | ^   oo . Let   (1)  (2)      (n)
X n ,X n ,...,X n  be a rearrangement of X1,...,Xn  such that   (1)     (2)         (n)
|X n |> |Xn |> ...> |Xn  |. Consider the sequence of weighted sums       sum n
Tn =   i=1 aiXi,n > 1, and, for fixed r > 1, set        sum 
Tn(r)=   ni=1aiXiI(| Xi|< |X(rn+1)|),n > r + 1; i.e., T(nr)  is the sum Tn  minus the sum of the X(kn)  ’s multiplied by their corresponding coefficients for k = 1,...,r. The main results provide sufficient and, separately, necessary conditions for b-1Tn(r)- kn --> 0
 n almost surely for some sequence of centering constants (kn,n > 1). The current work extends that of Mori [14], [15] wherein an  =_  1.

2000 AMS Mathematics Subject Classification: 60F1S.

Key words and phrases: Extreme terms, lightly trimmed sums, almost sure convergence, strong law of large numbers.

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