THE ASYMPTOTIC-DISTRIBUTION OF INTERMEDIATE SUMS

Let X(1,n) less-than-or-equal-to ... less-than-or-equal-to X(n,n) be t he order statistics of n independent random variables with a common di stribution function F and let k(n) be positive numbers such that k(n) --> infinity and k(n)/n --> 0 as n --> infinity, and consider the sums I(n)(a, b) = SIGMA(i=[akn]+1)[bkn]X(n+1-i,n) of intermediate order st atistics, where 0 < a < b. We find necessary and sufficient conditions for the existence of constants A(n) > 0 arid C(n) such that A(n)-1(I( n)(a,b)-C(n)) converges in distribution along subsequences of the posi tive integers {n} to nondegenerate limits and completely describe the possible subsequential limiting distributions.