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 n th record time and the n th record value, respectively. We refer to the variables X i as near-nth-record observations if X i ∈( 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.