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 = Sigma(n)(i=1) X-i, S-n(a) = Sigma(n)(i=1) XiI(M-n - a < X-i <= M-n) and V-n(2) = Sigma(n)(i=1) (X-i - X)(2) , where M-n = max(1 <= i <= n) X-i, X= (1/n) Sigma(n)(i=1) X-i, and a > 0 is a fixed constant. Under some suitable conditions, we show that (Pi([nt])(k=1) (T-k (a) / mu k))(mu/Vn) (sic) exp {integral(t)(0) (W(x) / x)dx} in D[0,1], as n -> infinity, where T-k(a) = S-k - S-k (a) is the trimmed sum and {W(t); t >= 0} is a standard Wiener process.