NEAR-EXTREMES AND RELATED POINT PROCESSES

Abstract

Let $X_i,i\ge 1$ be a sequence of random variables with continuous distribution functions and let $\left\{N\right(t),t\ge 0\}$ be a random counting process. Denote by $X_{i:N\left(t\right)},i\le N\left(t\right)$ the $i$-th lower order statistics of $X_1,\dots ,X_{N\left(t\right)},t\ge 0$ and define a point process in $\mathrm{ℝ}$ by ${\mathbf{M}}_{t,m}(·):=\sum _{i=1}^{N\left(t\right)}\mathbf{1}(X_{N\left(t\right)-m+1:N\left(t\right)}-X_i\in ·),m\in \mathrm{\mathbb{N}}$. In this paper we derive distributional and asymptotical results for ${\mathbf{M}}_{t,m}(·)$. For special marginals of the point process we retrieve some general results for the number of $m$-th near-extremes.