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 $\Phi(2u) \leqq c\Phi(u)$.
Citation
Adriano M. Garsia. "On a Convex Function Inequality for Martingales." Ann. Probab. 1 (1) 171 - 174, February, 1973. https://doi.org/10.1214/aop/1176997032
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