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WROCŁAW UNIVERSITY
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TECHNOLOGY

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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