Exact and asymptotic properties of delta-records in the linear drift model

Abstract

The study of records in the linear drift model (LDM) has attracted much attention recently due to applications in several fields. In the present paper we study delta-records in the LDM, defined as observations which are greater than all previous observations, plus a fixed real quantity delta. We give analytical properties of the probability of delta-records and study the correlation between delta-record events. We also analyse the asymptotic behaviour of the number of delta-records among the firstnobservations and give conditions for convergence to the Gaussian distribution. As a consequence of our results, we solve a conjecture posed inJ. Stat. Mech.2010P10013, regarding the total number of records in an LDM with negative drift. Examples of application to particular distributions, such as Gumbel or Pareto are also provided. We illustrate our results with a real data set of summer temperatures in Spain, where the LDM is consistent with the global-warming phenomenon.