Uniform Inequalities for Conditional Expectations

Abstract

The purpose of this note is to show that Neveu's uniform inequality for conditional expectations can be sharpened and extended to arbitrary conditioning sub-$\sigma$-fields. An application of this inequality yields that a sequence of conditional expectations given a $\sigma$-field $\mathscr{F}_n$ converges uniformly for all test functions to a conditional expectation given a $\sigma$-field $\mathscr{F}_\infty$ if and only if the $\sigma$-fields $\mathscr{F}_n$ converge to $\mathscr{F}_\infty$ in the usual metric.