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.

中文翻译：

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线性漂移模型中δ记录的精确和渐近性质

由于在多个领域的应用，线性漂移模型（LDM）中的记录研究最近引起了广泛关注。在本文中，我们研究 LDM 中的 $\delta$-记录，定义为大于所有先前观察的观察值，加上固定的实际数量 $\delta$。我们给出了 $\delta$-records 概率的分析特性，并研究了 $\delta$-record 事件之间的相关性。我们还分析了第一个 $n$ 观测值中 $\delta$ 记录数量的渐近行为，并给出了收敛到高斯分布的条件。根据我们的结果，我们解决了 J. Stat 中提出的一个猜想。机械。2010 年，P10013，关于 LDM 中具有负漂移的记录总数。还提供了应用于特定分布的示例，例如 Gumbel 或 Pareto。