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Some Limit Theorems for Extremal and Union Shot‐Noise Processes
Author(s) -
Heinrich L.,
Molchanov I. S.
Publication year - 1994
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19941680109
Subject(s) - mathematics , limit (mathematics) , shot noise , weak convergence , noise (video) , sequence (biology) , law of large numbers , limit point , mathematical analysis , infinity , function (biology) , limiting , pure mathematics , space (punctuation) , point process , discrete mathematics , image (mathematics) , random variable , statistics , computer science , artificial intelligence , detector , mechanical engineering , linguistics , philosophy , electrical engineering , computer security , evolutionary biology , biology , engineering , asset (computer security) , genetics
We study the limiting behaviour of suitably normalized union shot‐noise processes, where F is a set‐valued function on R d × ℳ is a sequence of i.i.d. random elements on some measurable space [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.