On the convergence of point processes
type: Item
instance of: scholarly article
title: On the convergence of point processes
zbMATH Open document ID: 0652.60054
DOI: 10.1007/BF00960075
author: Jean Jacod
publication date: 1987-01-01T00:00:00Z
review text: Let \(N^n\) be a counting process with compensator \(A^n\), \(n\in \mathbb N\), and \(N\) a counting process with compensator \(A=A(N)\). It is well-known that (under certain continuity assumptions on \(A\)) the condition \[ \text{(a)}\quad P-\lim_n(A^n_t-A_t(N^n))=0\quad \text{ for all } t\text{ implies } \text{(b)}\quad N^n\to N\quad (\text{in law}) \] but (b) does not imply (a), even for \(A_ t=t\). It is shown here that (b) is equivalent to another condition on convergence of the compensators. This looks much more complicated and involves weak convergence in \(L^1\) of their values in certain stopping times.
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 4064174
zbMATH Keywords: point processes
zbMATH Keywords: convergence in law
zbMATH Keywords: counting process with compensator
zbMATH Keywords: convergence of the compensators
zbMATH Keywords: weak convergence
zbMATH Keywords: stopping times
reviewed by: Yuri Kabanov
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4064174