Abstract
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.
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References
Chentsov, N.N. (1956): Weak convergence of stochastic processes whose trajectories have no discontinuities of the second kind and the heuristic approach to the Kolmogorov-Smirnov tests. Theor. Probability Appl. 1 (1956), 140–144.
Cramer, H. and Leadbetter, M.R. (1967): Stationary and related stochastic processes. Wiley, New York (1967).
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.
Gaenssler, P. and Stute, W. (1977): Wahrscheinlichkeitstheorie. Springer, Berlin-Heidelberg-New York (1977).
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).
Hahn, M.G. and Klass, M.J. (1977): Sample continuity of square-integrable processes. Ann. Probability 5 (1977), No. 3, 361–370.
Loève, M. (1963): Probability theory. 3rd edition, van Nostrand, Princeton (1963).
Mann, H.B. (1951): On the realization of stochastic processes by probability distributions in function spaces. Sankhyā Ser. A 11 (1951), 3–8.
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© 1978 ACADEMIA, Publishing House of the Czechoslovak Academy of Sciences, Prague
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Gaenssler, P., Stute, W. (1978). On Realizability of Stochastic Processes. In: Transactions of the Eighth Prague Conference. Czechoslovak Academy of Sciences, vol 8A. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9857-5_21
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DOI: https://doi.org/10.1007/978-94-009-9857-5_21
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