A COMPLETE SOLUTION FOR WEAK-CONVERGENCE OF HEAVILY TRIMMED SUMS

Let {X(n), n greater-than-or-equal-to 1} be a sequence of iidrvs with a common df F and for every n, let X(n,1) less-than-or-equal-to ... less-than-or-equal-to X(n,n) denote the order statistics of X1, ..., X(n). Consider the sums [GRAPHICS] where k(n) and l(n) satisfy k(n)/(n + l)-->a and l(n)/(n + 1)-->b for some 0<a<b<1. This paper gives necessary and sufficient conditions for (Sn* - (n + 1) M(k(n)/(n + 1), l(n)/(n + 1))/(n + 1)1/2 to converge weakly to a df G, where M(s, t) = integral-t/s F-(w)dw for 0<s<t<1; F-(t) = inf{x:F(x)greater-than-or-equal-to t}.