## The Annals of Mathematical Statistics

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

### A Generalization of Separable Stochastic Processes

#### 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