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