The Annals of Probability

On a Convex Function Inequality for Martingales

Adriano M. Garsia

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60G45

Keywords
26-A51 Martingales convex convex function inequalities for martingales

Citation

Garsia, Adriano M. On a Convex Function Inequality for Martingales. Ann. Probab. 1 (1973), no. 1, 171--174. doi:10.1214/aop/1176997032. https://projecteuclid.org/euclid.aop/1176997032


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