Limit theorems for the number and sum of near-maxima for medium tails

Let X1,X2,..., be a sequence of i.i.d. random variables. Xj, j[less-than-or-equals, slant]n is called a near-maximum iff Xj falls within a distance of the maximum Mn=max{X1,...,Xn}. In this paper, we focus on medium tailed distributions. A useful relationship on the number of near-maxima is built between general medium tailed and exponential distributions. Limit properties of the ratio Sn(a)/Sn are discussed, where Sn(a) is the sum of near-maxima.