Central Limit Theorem for the Number of Near-Records

摘要

Near-records in a sequence of random variables X_n,n≥1, are observations within a fixed distance of the current maximum. More precisely, as defined by Balakrishnan et al. (2005), X_n is a near-record if X_n ∈ (M_(n-1) - a, M_(n-1)], where M_n = max{X_1,..., X_n) and a>0 is fixed. In this article we establish the asymptotic normality of D_n = ∑_(i=1)~n 1_({x_i∈(M_(i=1) -a,M_(i=1)]}), the number of near-records among the first n observations, when the underlying random variables are independent and identically distributed, with common continuous distribution.