Some Limit Theorems for Extremal and Union Shot-Noise Processes
✍ Scribed by L. Heinrich; I. S. Molchanov
- Publisher
- John Wiley and Sons
- Year
- 2006
- Tongue
- English
- Weight
- 900 KB
- Volume
- 168
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
We study the limiting behaviour of suitably normalized union shot‐noise processes magnified image, where F is a set‐valued function on R^d^ × ℳ︁ is a sequence of i.i.d. random elements on some measurable space [ℳ︁ 𝔐] and Ψ = {x~i~, i≥ 1} stands for a stationary d‐dimensional point process whose intensity λ tends to infinity. General results concerning weak convergence of parametrized union shot‐noise processes Ξϵ(t) as ϵ ↓ 0 are obtained (Theorem 5.1 and its corollaries), if the point process λ^1 d^Ψ has a weak limit and F satisfies some technical conditions. An essential tool for proving these results is the notion of regular variation of multivalued functions. Some examples illustrate the applicability of the results.