Exceedances over upper order statistics in the linear drift model

M. Lafuente Blasco, M. Alcalde Navarro, F. J. López Lorente, G. Sanz Sáiz

We study exceedances over upper order statistics in the linear drift model $Y_n=X_n+cn$. In particular, we define $\delta$-$k$-records as observations exceeding the previous $k$-th upper order statistic by more than a margin $\delta$, positive or negative. The case $k=1$, $\delta=0$ recovers classical record results in the linear drift model, while the case $k=1$, $\delta\neq 0$ corresponds to the previously studied $\delta$-record setting. For $k\neq 1$, we derive the exact finite-sample probability through Poisson-binomial lower tails, prove affine invariance, monotonicity, continuity and positivity of the limiting probability, classify almost-sure finiteness, and establish a law of large numbers and central limit theorems in bounded- and unbounded-support regimes. We also give exact explicit examples and illustrations of our results.

Keywords: order statistics, records, linear trend, asymptotic theory

GT Procesos Estocásticos y sus Aplicaciones I
September 2, 2026  11:20 AM
Aula 26

M. Alcalde Navarro, R. Gouet Bañares, M. Lafuente Blasco, F. J. López Lorente, G. Sanz Sáiz

J. A. Vega Coso

J. Vives Santa-Eulalia