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 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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