## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 42, Number 5 (1971), 1647-1655.

### Some Conditions Under Which Two Random Variables are Equal Almost Surely and a Simple Proof of a Theorem of Chung and Fuchs

#### Abstract

Let $(X, Y)$ be an ordered pair of real-valued random variables. Say that $(X, Y)$ is fair if $E(Y \mid X) = X$ a.s. It is shown, for example, that if $X$ has a finite mean and the pair $(X, Y)$ is fair, then $X$ and $Y$ cannot be stochastically ordered unless $X = Y$ a.s. The conclusion is in general false, if $X$ does not have a mean. On the other hand, if $X$ is independent of the increment $Y - X$, the preceding statement remains in force without any moment restrictions on $X$. The last assertion, combined with a gambling idea of Dubins and Savage, yields a simple proof of a theorem of Chung and Fuchs on the upper limit of a random walk with mean zero.

#### Article information

**Source**

Ann. Math. Statist., Volume 42, Number 5 (1971), 1647-1655.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177693163

**Digital Object Identifier**

doi:10.1214/aoms/1177693163

**Mathematical Reviews number (MathSciNet)**

MR346898

**Zentralblatt MATH identifier**

0239.60004

**JSTOR**

links.jstor.org

#### Citation

Gilat, David. Some Conditions Under Which Two Random Variables are Equal Almost Surely and a Simple Proof of a Theorem of Chung and Fuchs. Ann. Math. Statist. 42 (1971), no. 5, 1647--1655. doi:10.1214/aoms/1177693163. https://projecteuclid.org/euclid.aoms/1177693163