On the asymptotic behavior of mixed Poisson processes
type: Item
instance of: scholarly article
title: On the asymptotic behavior of mixed Poisson processes
zbMATH Open document ID: 0911.60033
DOI: 10.1007/BF02986859
published in: Lithuanian Mathematical Journal
publication date: 1998-04-01T00:00:00Z
review text: A mixed Poisson process \(N(t)\) \((t>0)\) is considered which is a Poisson process with a random intensity \(\lambda\). The following formula is proved \[ \limsup_{t\to\infty}{N(t)-A(t)\over\sqrt{2t\log\log t}}=\sqrt\lambda, \] where \(A(t)\) is a compensator of the process \(N\) such that \(M(t)=N(t)-A(t)\) is a martingale. A transformation of an \(N\)-predictable process \(\varphi\) of the form \[ X_t(\varphi)=\int^t_0\varphi(s)dM(s)\quad(t\geq 0) \] is investigated. A law of iterated logarithms is proved for this process as \(t\to\infty\) and also an asymptotics of a variance of \(X_t(\varphi)\) is derived under assumption \(\varphi\) increases as \(t^\gamma\) \((\gamma\in[0,\infty))\).
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 1135556
zbMATH Keywords: martingale
zbMATH Keywords: iterated logarithm
zbMATH Keywords: stochastic integral
zbMATH Keywords: compensator
author: Bronius Grigelionis
reviewed by: B. P. Harlamov
MaRDI profile type: Publication
cites work: Q4229047
cites work: On mixed poisson processes and martingales
cites work: Q4155574
cites work: A strong law of large numbers for local martingales
cites work: A law of the iterated logarithm for stochastic integrals
cites work: Q4098439
cites work: Q5509983
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description: scientific article; zbMATH DE number 1135556