A limit theorem for random products of trimmed sums of i.i.d. random variables

Let { x,x i; i ≥ 1 } be a sequence of independent and identically distributed positive random variables with a continuous distribution function f,and f has a medium tail. denote s n = ∑ i=1 n x i,s n (a) = ∑ i=1 n x i i (m n - a x i < m n) and v n 2 = ∑ i=1 n (x i - x ) 2,where m n = max 1<i<n x i,x = (1/n) ∑ i=1 n x i,and a > 0 is a fixed constant. under some suitable conditions,we show that (π k=1 [nt] (t k (a) μ/k)) μ/v n → d exp {∫ 0 t (w(x)/x)dx} in d [0,1],as n →,where t k (a) = s k - s k (a) is the trimmed sum and { w (t); t 0 } is a standard wiener process. copyright © 2011 fa-mei zheng.