Doob's martingale convergence theorems

As Martingalkonvergenzsatz certain statements about the convergence of martingales are called in probability theory. A martingale is a special stochastic process, which can be regarded as a formalization and generalization of fair gaming. Under additional assumptions on the boundedness of the process, the convergence can be inferred. Here, the different versions of the sentence with regard to the nature of the limitations and the type of convergence differ.

Requirements

On a probability space with a filtration and is a sequence of real random variables given, which is adapted to the filtration and is integrable. This means that for all the random variables can be measured with respect to and fulfilled.

The process is called a martingale if the equation is valid for all. Applies instead for all then the process is called a submartingale. In the case of all the process is called a supermartingale. Every martingale is a sub and a supermartingale. A process is a supermartingale if and only if a submartingale is.

Versions of the Martingalkonvergenzsatzes

Almost sure convergence

It is a submartingale and there is a constant with all that is, the expected value of the positive parts is limited. Then a - measurable random variable exists with almost certain.

Convergence in pth mean

Be and there is a constant with all that is, the sequence is bounded in the space then an - measurable random variable exists almost surely and in.

The statement is false for in general: One in bounded martingale does not necessarily have to converge.

Convergence in gleichgradiger integrability

If a is uniformly integrable submartingale, then there exists a - measurable random variable with almost surely and in.

Next is valid and, in the case that is a martingale, even. It is said that the martingale is completed by.