Doob's martingale inequality

Doob's martingale inequalities bound the maximum of a submartingale by its terminal value, in probability or in moments. They are fundamental tools for controlling fluctuations over time.

Submartingale: A stochastic process whose conditional expected next value is at least its current value. The inequality applies to submartingales and exploits their upward conditional drift.
Filtration (probability theory): An increasing sequence of sigma-algebras representing the information available over time. A martingale inequality is formulated relative to an evolving information structure.
Martingale convergence theorem: A theorem giving conditions under which a martingale converges almost surely or in an integrable norm. Maximal bounds help control oscillations while proving martingale convergence.
Doob's Lp inequality: The bound stating that the Lp norm of a submartingale's maximum is at most p/(p−1) times its terminal Lp norm, for p greater than one. This is the moment version of Doob's maximal control.
Stopping time: A random time whose occurrence by each moment can be determined from information available up to that moment. Stopping at the first threshold crossing turns a maximum event into a terminal-value bound.
Martingale: An integrable stochastic process whose conditional expected future value equals its present value. Martingales are the central special case of the submartingales covered by Doob's bounds.
Uniform integrability: A condition ensuring that a family of random variables has uniformly small expectation tails. Doob's maximal inequality helps transfer uniform integrability from terminal values to process maxima.
Doob's weak L1 inequality: The bound that the probability a nonnegative submartingale's maximum exceeds a positive threshold is at most its terminal expectation divided by that threshold. This is the probability version, requiring only a first-moment terminal bound.
First-passage time: The earliest time a stochastic process reaches or crosses a specified level. The maximum event is expressed through the first time the process crosses a threshold.
Maximum (stochastic process): The largest value attained by a stochastic process over a specified time interval. The inequality controls this pathwise maximum using information at the terminal time.

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