Sharp moment estimates for martingales with uniformly bounded square functions

Abstract

We provide sharp bounds for the exponential moments and $p$-moments, $1⩽p⩽2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on $\mathrm{BMO}([0,1])$. In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.

33 pages, 4 figures; second version corrects some typos

References in corpus (1)

• Sharp constants in the classical weak form of the John--Nirenberg inequality

Cited by in corpus (1)

• Exponential integrability for log-concave measures