A functional extremal criterion
type: Item
instance of: scholarly article
title: A functional extremal criterion
zbMATH Open document ID: 1074.60056
author: Elisaveta Pancheva
author: Pavlina Jordanova
published in: Journal of Mathematical Sciences (New York)
publication date: 2005-06-28T00:00:00Z
review text: Let \({\mathcal N}=\{(t_k, X_k):\, k\geq 1\}\) be a point process with time space \([0, \infty)\) and state space \([0, \infty)^d\), where \(\{t_k\}\) are distinct nonrandom time points monotonically increasing to \(\infty\). \(\{X_k\}\) are independent and identically distributed random vectors on a given probability space with values in \([0,\infty)^d\) and with common distribution function nondefective at \(+\infty\). Assume that almost all realizations of \({\mathcal N}\) are Radon measures on \({\mathcal S}=[0,\infty)\times E\), where \(E=[0,\infty]^d\setminus\{0\}\). The authors consider two random processes associated with \({\mathcal N}\), namely, the extremal process \(X(t) =\{\vee X_k:\, t_k\leq t\}\) and the process \(Z(t)= \{\sum X_k:\, t_k\leq t\}\) with additive increments. Denote by \({\mathcal M}\) the set of all nondecreasing, right-continuous functions \(y: (0,\infty) \to [0, \infty)^d\). Then the set \({\mathcal P}\) of all probability measures on \({\mathcal M}\) is compact. Let \(\{P_n\}\) be a sequence of probability measures on \({\mathcal M}\). \(\{P_n\}\) is weakly convergent to \(P\in {\mathcal P}\), briefly \(P_n \Rightarrow P\), if \(\int \varphi dP_n \to \int \varphi dP\) for bounded \(\varphi: {\mathcal M} \to R\) which are continuous in the weak topology of \({\mathcal M}\). Denote by \({\mathcal P}_e\) and \({\mathcal P}_s\) the subsets of \({\mathcal P}\) corresponding to an extremal process (with independent max-increments) and to a sum process (with independent additive increments), respectively. \textit{A. A. Balkema} and \textit{E. I. Pancheva} [Commun. Stat., Theory Methods 25, No. 4, 737--758 (1996; Zbl 0875.60010), Theorem 6.4] have shown that the space \({\mathcal P}_e\) with the topology of weak convergence is closed in \({\mathcal P}\). The same is also true for \({\mathcal P}_s\). So, the weak convergence of extremal processes \(Y_n\Rightarrow Y\) and of sum processes \(S_n \Rightarrow S\) is equivalent to the convergences…
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G18
zbMATH DE Number: 2183273
zbMATH Keywords: point processes
zbMATH Keywords: extremal processes
zbMATH Keywords: weak convergence
zbMATH Keywords: self-similar processes
reviewed by: Viktor Ohanyan
MaRDI profile type: Publication
DOI: 10.1023/B:JOTH.0000027029.93702.4E
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description: scientific article; zbMATH DE number 2183273