= Dubins upcrossing inequality
{c}
{title2=$\mathbb P(U\geq k)\leq(a/b)^k\mathbb E\min(X_0/a,1)$}
= Dubins inequality
{c}
{synonym}
For a nonnegative <supermartingale> and $0<a<b$, let $U$ count completed <upcrossings> of $[a,b]$, allowing the first entry at time zero. The displayed bound holds for every integer $k\geq1$. Construct the <multiplicative upcrossing supermartingale>, stop at the $k$th completed crossing, and use its initial <expected value> and nonnegativity. The bound is exponential in the number of crossings, and complements the first-moment <Doob upcrossing inequality>.
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