On Realizability of Stochastic Processes
Let ξ = (ξt)0≤t≤1 be a stochastic process on some probability space (Ω, ℱ ℙ) and let X be a fixed class of real functions on the unit interval. Then, by definition, ξ is said to have a realization in X, if there exists a process n = (ηt)0≤t≤1 with sample paths in X and such that ξ and η have the same finite dimensional distributions. It is the purpose of this paper to point out the strong interdependence between realizability in X and the behaviour of a corresponding modulus function.
KeywordsSample Path Unit Interval Modulus Function Nondecreasing Function Fixed Class
Unable to display preview. Download preview PDF.
- Gaenssler, P. (1974): On the realization of stochastic processes by probability distributions in function spaces. To appear in: Trans, of the Seventh Prague Conference, Prague 1974.Google Scholar
- Gihman, I.I. and Skorohod, A.V. (1974): The theory of stochastic processes I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 210, Springer, Berlin-Heidelberg-New York (1974).Google Scholar
- Loève, M. (1963): Probability theory. 3rd edition, van Nostrand, Princeton (1963).Google Scholar