 Maximal bound for a nonnegative martingale

ID: maximal-bound-for-a-nonnegative-martingale

For a nonnegative martingale,

T_{R} = in f {n : M_{n} \geq R}

satisfies

P (T_{R} < \infty) \leq E M_{0} / R

. Stop at

T_{R} \land n

and use nonnegativity before passing to the increasing event. A limit of thresholds from below also gives the bound for

{sup_{n} M_{n} \geq R}

even when the supremum is not attained.

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