The number and sum of near-maxima for thin-tailed populations

A near-maximum is an observation which falls within a distance a of the max imum observation in an i.i.d. sample of size n. The asymptotic behaviour of the number K-n(a) of near-maxima is known for the cases where the right ex tremity of the population distribution function is finite, and where it is infinite and the right hand tail is exponentially small, or fatter than exp onential. This paper completes the picture for thin tails, i.e., tails whic h decay faster than exponential. Limit theorems are derived and used to fin d the large-sample behaviour of the sum of near-maxima.