On the number and sum of near-record observations

Let X-1, X-2,... be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L (n) and X (n) denote the nth record time and the nth record value, respectively. We refer to the variables X-i as near-nth-record observations if X-i epsilon (X (n) - a, X (n)], with a > 0, and L (n) < i < L (n + 1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.