Limit Theorems for the Counting Process of Near-Records

摘要

Near-records of a sequence, as defined in Balakrishnan et at. (2005), are observations lying within a fixed distance of the current record. In this article we study the asymptotic behavior of the number of near-records, among the first n observations in a sequence of independent, identically distributed and absolutely continuous random variables. We give conditions for the finiteness of the total number of near-records as well as laws of targe numbers for their counting process. For distributions with a finite number of near-records, we carry out a simulation study suggesting that the total number of near-records has a geometric distribution.