### Near-extremes and Related Point Processes

*Enkelejd Hashorva*

#### Abstract

Let $X_i,i\ge 1$ be a sequence of random variables

with continuous distribution functions and let $\{N(t), t \ge 0\}$

be a random counting process. Denote by $X_{i:N(t)}, i\le N(t)$ the

$i$-th lower order statistics of $X_1 \ldot X_{N(t)}, t\ge 0$ and

define a point process in $\R$ by $\kal{M}_{t,m}(\cdot):

=\sum_{i=1}^{N(t)}\vk{1}(\ORD{X}{m}- X_i \in \cdot),m\inn$. In this paper

we derive distributional and asymptotical results for $\kal{M}_{t,m}(\cdot)$. For special marginals of the

point process we retrieve some general results for the number of $m$-th near-extremes.

with continuous distribution functions and let $\{N(t), t \ge 0\}$

be a random counting process. Denote by $X_{i:N(t)}, i\le N(t)$ the

$i$-th lower order statistics of $X_1 \ldot X_{N(t)}, t\ge 0$ and

define a point process in $\R$ by $\kal{M}_{t,m}(\cdot):

=\sum_{i=1}^{N(t)}\vk{1}(\ORD{X}{m}- X_i \in \cdot),m\inn$. In this paper

we derive distributional and asymptotical results for $\kal{M}_{t,m}(\cdot)$. For special marginals of the

point process we retrieve some general results for the number of $m$-th near-extremes.

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