A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process.

The central limit problem for trimmed sums

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.