Exceedances over upper order statistics in the linear drift model
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.
Palabras clave: order statistics records linear trend asymptotic theory