Central limit theorem for homogeneous processes with independent increments and semi-Markov commutations
type: Item
instance of: scholarly article
title: Central limit theorem for homogeneous processes with independent increments and semi-Markov commutations
zbMATH Open document ID: 0539.60089
DOI: 10.1007/BF01056232
author: V. V. Korolyuk
author: V. S. Korolyuk
published in: Ukrainian Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: A mixed process with independent increments \((Y(t))_{t\geq 0}\) is considered. The cumulant of the process is a stepped semi-Markov process. The family \((\alpha_{\epsilon})_{\epsilon>0}\) of derived processes is investigated where \(\alpha_{\epsilon}(t)=Y(t/\epsilon)\sqrt{\epsilon}- \Lambda t\sqrt{\epsilon}\), \(\Lambda =m^{- 1}\int_{E}m(x)\lambda(x)\rho(dx)\), \(m=\int_{E}m(x)\rho(dx)\), \(m(x)=\int^{\infty}_{0}tG_ x(dt)\), \(G_ x\) is the distribution function of the first exit time from the state \(x\in E\) of the control semi-Markov process, \(\rho\) is the stationary distribution of the embedded Markov chain and \(\lambda\) (x) is the conditional cumulant of the process Y given the state x of the control semi-Markov process. Some conditions in terms of the control semi-Markov process and conditional cumulants are proved to be sufficient for finite dimensional distributions of the limit process \(\alpha =\lim_{\epsilon \to 0}\alpha_{\epsilon}\) to be normal.
reviewed by: B. P. Harlamov
Mathematics Subject Classification ID: 60K15
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 3858134
zbMATH Keywords: semi-Markov process
zbMATH Keywords: first exit time
MaRDI profile type: Publication
cites work: Q3245609
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description: scientific article; zbMATH DE number 3858134