Doob's martingale convergence theorems

In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences.

Statement for discrete-time martingales

A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let

X

1

,

X

2

,

X

3

,

…

{\displaystyle X_{1},X_{2},X_{3},\dots }

be a supermartingale. Suppose that the supermartingale is bounded in the sense that

sup

t

∈

N

E

⁡

[

X

t

−

]

<

∞

{\displaystyle \sup _{t\in \mathbf {N} }\operatorname {E} [X_{t}^{-}]<\infty }

where

X

t

−

{\displaystyle X_{t}^{-}}

is the negative part of

X

t

{\displaystyle X_{t}}

, defined by

X

t

−

=

−

min

(

X

t

,

0

)

{\textstyle X_{t}^{-}=-\min(X_{t},0)}

. Then the sequence converges almost surely to a random variable

X

{\displaystyle X}

with finite expectation.

There is a symmetric statement for submartingales with bounded expectation of the positive part. A supermartingale is a stochastic analogue of a non-increasing sequence, and the condition of the theorem is analogous to the condition in the monotone convergence theorem that the sequence be bounded from below.