The Annals of Mathematical Statistics

A Generalization of Separable Stochastic Processes

E. O. Elliott

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Abstract

Doob introduced the standard modifications or extensions of a stochastic process and proved that every stochastic process has a separable standard modification. In 1964 Elliott and Morse developed a general theory of product measures with implications in the theory of continuous parameter processes with mutually independent random variables. In particular, they gave a new method for obtaining extensions which considerably generalizes the notion of separability. For a separable process only certain events specified by restrictions of the random variables at a nondenumerable collection of time points are measurable. Under their generalization, the restriction to only certain events is virtually removed. The key to the new method for obtaining extensions is a modification by means of nilsets. The definition of nilsets has recently been adjusted to enable the application of this method to general stochastic processes.

Article information

Source
Ann. Math. Statist., Volume 43, Number 1 (1972), 320-325.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177692725

Digital Object Identifier
doi:10.1214/aoms/1177692725

Mathematical Reviews number (MathSciNet)
MR319251

Zentralblatt MATH identifier
0241.60026

JSTOR
links.jstor.org

Citation

Elliott, E. O. A Generalization of Separable Stochastic Processes. Ann. Math. Statist. 43 (1972), no. 1, 320--325. doi:10.1214/aoms/1177692725. https://projecteuclid.org/euclid.aoms/1177692725


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