Ville inequality
= Ville inequality
{c}
{title2=$\Pr(\sup_t M_t\geq a)\leq\mathbb EM_0/a$}
For a nonnegative <martingale>, the displayed maximal probability bound follows by stopping at the first crossing of $a$. The stopped expectation is at most its initial expectation, and on the crossing event its value is at least $a$. Bound the stopping time first and pass to the limit using nonnegativity. This form is useful for the <Poisson maximal concentration bound>.