Limit Theorems for the Counting Process of Near-Records

Abstract

Near-records of a sequence, as defined in Balakrishnan et al. (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 large 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.

Keywords:

• Limit theorem
• Near-record

Mathematics Subject Classification:

• Primary 60F15
• Secondary 60G70

Acknowledgments

We thank the Editor and anonymous referees for careful reading of the manuscript and valuable suggestions. Support by FONDAP and BASAL-CMM projects, Fondecyt grant 1090216, and projects MTM2007-63769 of MEC and MTM2010-15972 of MICINN is gratefully acknowledged. The authors are members of the research group Modelos Estocásticos (DGA).