martingale convergence theorem
There are several convergence theorems for martingales, which follow from Doob’s upcrossing lemma. The following says that any -bounded martingale 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 is understood to be defined with respect to a probability space and filtration .
Theorem (Doob’s Forward Convergence Theorem).
Let be a martingale (or submartingale, or supermartingale) such that is bounded over all . Then, with probability one, the limit exists and is finite.
The condition that is -bounded is automatically satisfied in many cases. In particular, if is a non-negative supermartingale then for all , so is bounded, giving the following corollary.
Let be a non-negative martingale (or supermartingale). Then, with probability one, the limit exists and is finite.
Let be a standard random walk. That is, and
Then, for every integer , with probability one for some .
Without loss of generality, suppose that . Let be the first time for which . It is easy to see that the stopped process defined by is a martingale and is non-negative. Therefore, by the martingale convergence theorem, the limit exists and is finite (almost surely). In particular, converges to and must be less than for large . However, whenever , so we have and therefore for some . ∎
|Title||martingale convergence theorem|
|Date of creation||2013-03-22 18:33:47|
|Last modified on||2013-03-22 18:33:47|
|Last modified by||gel (22282)|