On the distribution of maxima of martingales

Abstract:

Partial order the set of distributions on the real line by $\nu \leqslant \nu ’$ if $\nu (x,\infty ) \leqslant \nu ’(x,\infty )$ for all x. Then, for each $\mu$ with a finite first moment, the family $M(\mu )$ of all $\nu$ which are distributions of (essential) suprema of martingales closed on the right by a $\mu$-distributed random number, has a least upper bound ${\mu ^ \ast }$, and is, therefore, a tight family. In fact, ${\mu ^ \ast }$ is $\bar \mu$, the distribution of the Hardy-Littlewood extremal maximal function associated with $\mu$. Moreover, ${\mu ^ \ast }$ is itself an element of $M(\mu )$. For each $p > 1$, the classical moment inequality that the ${L_p}$ norm of $\bar \mu$ (and of ${\mu ^ \ast }$) is at most $p/(p - 1)$ times the ${L_p}$ norm of $\mu$ is shown to be sharp.