Abstract
This chapter deals with almost sure continuity of sample paths of multi-dimensional Gaussian Markov processes. The well-known Kinney-Dynkin criterion of continuity of Markov processes is complemented by an entropy criterion which gives evidence that the necessary conditions of continuity of Gaussian processes due to V.Sudakov are sufficient for continuity of Gaussian Markov processes. This problem is considered in Section 6.4 which logically accomplishes the investigation started in Section 5.2. In Section 6.3, the link is established between the points of discontinuity of sample paths of a multi-dimensional Gaussian Markov process and the points where the rank of variance matrice of this process varies. The statements of Sections 6.3 and 6.4 dwell on the claim on equivalence of sequential and sample almost sure continuity of Gaussian processes. This claim complements the well-known oscillation theorem due to K.Itô and M.Nisio and reduces the problem of sample continuity of almost all sample paths of Gaussian processes to that of the almost sure convergence to zero of Gaussian sequences. Even though the statement on reduction is proved under fairly general assumptions on the parametric set and the range of values of the processes considered, the simplest situations in applications already prove the utility of this statement.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Buldygin, V., Solntsev, S. (1997). Continuity of sample paths of Gaussian Markov processes. In: Asymptotic Behaviour of Linearly Transformed Sums of Random Variables. Mathematics and Its Applications, vol 416. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5568-7_7
Download citation
DOI: https://doi.org/10.1007/978-94-011-5568-7_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6346-3
Online ISBN: 978-94-011-5568-7
eBook Packages: Springer Book Archive