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martingale convergence theorem∗
gel†
2013-03-22 2:04:43
There are several convergence theorems for martingales, 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∈N is understood to be defined with respect to a probability space (Ω, F, P) and filtration
(Fn )n∈N .
Theorem (Doob’s Forward Convergence Theorem). Let (Xn )n∈N be a martingale (or submartingale, or supermartingale) such that E[|Xn |] is bounded over
all n ∈ N. Then, with probability one, the limit X∞ = limn→∞ Xn 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 n ≥ 1, so E[|Xn |] is bounded, giving the following corollary.
Corollary. Let (Xn )n∈N be a non-negative martingale (or supermartingale).
Then, with probability one, the limit X∞ = limn→∞ Xn exists and is finite.
As an example application of the martingale convergence theorem, it is easy
to show that a standard random walk started started at 0 will visit every level
with probability one.
Corollary. Let (Xn )n∈N be a standard random walk. That is, X1 = 0 and
P(Xn+1 = Xn + 1 | Fn ) = P(Xn+1 = Xn − 1 | Fn ) = 1/2.
Then, for every integer a, with probability one Xn = a for some n.
Proof. Without loss of generality, suppose that a ≤ 0. Let T : Ω → N ∪ {∞}
be the first time n for which Xn = a. It is easy to see that the stopped process
∗ hMartingaleConvergenceTheoremi
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XnT defined by XnT = Xmin(n,T ) is a martingale and X T − a is non-negative.
T
Therefore, by the martingale convergence theorem, the limit X∞
= limn→∞ XnT
T
T
exists and is finite (almost surely). In particular, |Xn+1 − Xn | converges to 0
T
and must be less than 1 for large n. However, |Xn+1
−XnT | = 1 whenever n < T ,
so we have T < ∞ and therefore Xn = a for some n.
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