ON STABILITY OF TRIMMED SUMS
Tien-Chung Hu, Chiung‐Yu Huang, Andrew Rosalsky · 2003
Let {Xn, n ≥ 1} be a sequence of i.i.d. random variables and let {an, n ≥ 1} and {bn, n ≥ 1} be sequences of constants where 0 < bn ↑ ∞. Let X n , X (2) n , · · · , X n be a rearrangement of X1, · · · , Xn such that |X n | ≥ |X n | ≥ · · · ≥ |X n |. Consider the sequence of weighted sums Tn = ∑n i=1aiXi, n ≥ 1 and, for fixed r ≥ 1, set T (r) n = ∑n i=1aiXiI(|Xi| ≤ |X n |), n ≥ r+1; i.e., T (r) n is the sum Tn minus the sum of the X n ’s multiplied by their corresponding coefficients for k = 1, . . ., r. The main results provide sufficient and, separately, necessary conditions for b−1 n T (r) n −kn → 0 almost surely for some sequence of centering constants {kn, n ≥ 1}. The current work extends that of Mori [?, ?] wherein an ≡ 1. 2000 Mathematics Subject Classification: 60F15.