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.
Citation
E. O. Elliott. "A Generalization of Separable Stochastic Processes." Ann. Math. Statist. 43 (1) 320 - 325, February, 1972. https://doi.org/10.1214/aoms/1177692725
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