Central Limit Theorem for the Number of Near-Records

Abstract

Near-records in a sequence of random variables Xn n ≥ 1 are observations within a fixed distance of the current maximum. More precisely, as defined by Balakrishnan et al. (2005), Xn is a near-record if Xn ∈ Mn−1 − a Mn−1 , where Mn = max X1 Xn and a > 0 is fixed. In this article we establish the asymptotic normality of Dn = n i=1 1 Xi∈ Mi−1−a Mi−1 , the number of nearrecords among the first n observations, when the underlying random variables are independent and identically distributed, with common continuous distribution.