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 δ-records in the LDM, defined as observations which are greater than all previous observations, plus a fixed real quantity δ. We give analytical properties of the probability of δ-records and study the correlation between δ-record events. We also analyse the asymptotic behaviour of the number of δ-records among the first n observations and give conditions for convergence to the Gaussian distribution. As a consequence of our results, we solve a conjecture posed in J. Stat. Mech. 2010 P10013, 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.
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