I continue the theme of explaining important bookwork from the Part III course, Advanced Probability, as succinctly as possible. In this post, we consider the convergence properties of discrete time martingales.
1) Theorem: Assume is a martingale bounded in . Then
Remark: The theorem and proof works equally well for a supermartingale.
Proof: Essentially, we want to reduce convergence of the random variables which make up the martingale to a countable collection of events. We do this by considering upcrossings, which counts the number of times the process alternates from less than a given interval to greater than a given interval. The formal definition will be too wide for this format, so we summarise as
the number of disjoint time intervals up to time n in which X goes from to . Define to be the limit as n increases to infinity.
It is a genuinely easy check that a sequence converges (possibly to ) iff this number of upcrossings of any interval with rational bounds is finite. We will show that the martingale almost surely has this property. The key lemma is a bound due to Doob:
Proof: Say are the successive hitting times of respectively. So . We decompose, abbreviating the number of upcrossings as N.
Now take an expectation of both sides, applying the Optional Stopping Theorem to the bounded stopping times on the LHS. (If we are working with a supermartingale, formally we need to take of each summand on LHS to show that they are non-negative, and so taking a further expectation over gives the required result.) We obtain:
If then both . Otherwise . This complete the proof of the lemma.
Since , where this last bound is uniform in n by assumption, applying monotone convergence, we get that is almost surely finite for every pair . Because this set is countable, we can deduce that this holds almost surely for every pair simultaneously. We therefore define when this limit exists, and 0 otherwise. With probability one the limit exists. Fatou’s lemma confirms that .
2) We often want to have convergence in as well. Recall for Part II Probability and Measure (or elsewhere) that
UI + Convergence almost surely is necessary and sufficient for convergence in .
This applies equally well to this situation. Note that for a martingale, this condition is often convenient, because, for example, we know that the set is UI for any integrable .
3) Convergence in is easier to guarantee.
Theorem: i) X a martingale bounded in iff ii) in and almost surely iff iii) s.t. a.s.
Remark: As part of the proof, we will show, as expected, that are the same.
Proof: i)->ii) Almost sure convergence follows from the above result applied to the p-th power process. We apply Doob’s inequality about running maxima in a martingale process:
Using this, we see that . Now consider and use Dominated Convergence to confirm convergence in .
Note that Doob’s inequality can be proven using the same author’s Maximal inequality and Holder.
ii)->iii) As we suspected, we show is suitable.
iii)->i) is easy. Z bounded in implies X bounded by a simple application of the triangle inequality in the definition of conditional expectation.