## The Annals of Probability

### On a Convex Function Inequality for Martingales

#### Abstract

A new proof is given for the inequality $$E(\Phi(\Sigma^\infty_{\nu=1} E(z_\nu \mid \mathscr{F}_\nu))) \leqq CE(\Phi(\Sigma^\infty_{\nu=1} z_\nu)),$$ where $z_1, z_2, \cdots, z_n, \cdots$ are nonnegative random variables on a probability space $(\Omega, \mathscr{F}, \mathbf{P}), \mathscr{F}_1 \subset \mathscr{F}_2 \subset \cdots \subset \mathscr{F}_n \subset \cdots \mathscr{F}$ is a sequence of $\sigma$-fields and $\Phi(u)$ is a convex function satisfying $\Phi(2u) \leqq c\Phi(u)$.

#### Article information

Source
Ann. Probab., Volume 1, Number 1 (1973), 171-174.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176997032

Digital Object Identifier
doi:10.1214/aop/1176997032

Mathematical Reviews number (MathSciNet)
MR346897

Zentralblatt MATH identifier
0289.60026

JSTOR