In the theory of discrete time

Discrete time

Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...

 stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es, a part of the mathematical theory of probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, the Doob decomposition theorem gives a unique decomposition of any submartingale as the sum of a martingale

Martingale (probability theory)

In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

 and an increasing predictable process

Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process which the value is knowable at a prior time...

. The theorem was proved by and is named for J. L. Doob. The analogous theorem for continuous submartingales is the Doob–Meyer decomposition theorem.

The theorem

Any submartingale X_n has a unique decomposition X_n = M_n + A_n where M_n is a martingale and A_n is a predictable, increasing process with A₀ = 0.