martingale convergence theorem

There are several convergence theorems for martingalesMathworldPlanetmath, which follow from Doob’s upcrossing lemma. The following says that any L1-bounded martingale Xn in discrete time converges almost surely. Note that almost-sure convergence (i.e. convergence with probability one) is quite strong, implying the weaker property of convergence in probability. Here, a martingale (Xn)n is understood to be defined with respect to a probability spaceMathworldPlanetmath (Ω,,) and filtrationPlanetmathPlanetmath (n)n.

Theorem (Doob’s Forward Convergence Theorem).

Let (Xn)nN be a martingale (or submartingale, or supermartingale) such that E[|Xn|] is bounded over all nN. Then, with probability one, the limit X=limnXn exists and is finite.

The condition that Xn is L1-bounded is automatically satisfied in many cases. In particular, if X is a non-negative supermartingale then 𝔼[|Xn|]=𝔼[Xn]𝔼[X1] for all n1, so 𝔼[|Xn|] is bounded, giving the following corollary.


Let (Xn)nN be a non-negative martingale (or supermartingale). Then, with probability one, the limit X=limnXn exists and is finite.

As an example application of the martingale convergence theorem, it is easy to show that a standard random walkMathworldPlanetmath started started at 0 will visit every level with probability one.


Let (Xn)nN be a standard random walk. That is, X1=0 and


Then, for every integer a, with probability one Xn=a for some n.


Without loss of generality, suppose that a0. Let T:Ω{} be the first time n for which Xn=a. It is easy to see that the stopped process XnT defined by XnT=Xmin(n,T) is a martingale and XT-a is non-negative. Therefore, by the martingale convergence theorem, the limit XT=limnXnT exists and is finite (almost surely). In particular, |Xn+1T-XnT| converges to 0 and must be less than 1 for large n. However, |Xn+1T-XnT|=1 whenever n<T, so we have T< and therefore Xn=a for some n. ∎

Title martingale convergence theorem
Canonical name MartingaleConvergenceTheorem
Date of creation 2013-03-22 18:33:47
Last modified on 2013-03-22 18:33:47
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 60G46
Classification msc 60G44
Classification msc 60G42
Classification msc 60F15
Related topic Martingale
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Related topic UpcrossingsAndDowncrossings