Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 3, No 2 (2009)

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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.

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