δ -records in dependent sequences of random variables
Let $\delta$ be a real number and $\{X_j\}_{j\ge 1}$ be a sequence of random variables. An observation $X_n$ is said to be a $\delta$-record if
$$
X_n>\max\{X_1,\ldots,X_{n-1}\} + \delta.
$$
This generalization of the concept of record observation is especially interesting when $\delta <0$ as $\delta$-records show up more frequently. We analyze the well-known Linear Drift Model (LDM) $Y_n = X_n + cn$, $n\ge 1$, assuming that the residuals $\{X_n\}_{n\ge 1}$ are strongly stationary and strong mixing. Our purpose is to study the asymptotic behaviour of the sample $\delta$-record rate $N_{n,\delta}/n$. A Central Limit Theorem and a Law of Iterated Logarithm are proven. Finally, all these results are applied on a LDM with AR$(1)$ residuals.
Palabras clave: Records δ-Records Asymptotics Central Limit Theorem Law of Iterated Logarithm