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Navigation menu Personal tools Not logged in Talk Contributions Create account Log in. Namespaces Article Talk. Views Read Edit View history. Main page Contents Current events Random article About Wikipedia Contact us Donate. Help Learn to edit Community portal Recent changes Upload file. Example 1. Example 2. In the gambling world such a system is called a martingale, which explains the origin of the mathematical term "martingale". As a particular case of this the Wald identity follows:. are martingales.

which is a local martingale. In the case of continuous time the Doob, Burkholder and Davis inequalities are still true for right-continuous processes having left limits. Stopping times are also called optimal times, or, in the older literature, Markov times or Markov moments, cf. Markov moment. The optimal sampling theorem is also called the stopping theorem or Doob's stopping theorem. The notion of a martingale is one of the most important concepts in modern probability theory.

It is basic in the theories of Markov processes and stochastic integrals, and is useful in many parts of analysis convergence theorems in ergodic theory, derivatives and lifting in measure theory, inequalities in the theory of singular integrals, etc. Log in. Navigation Main page Pages A-Z StatProb Collection Recent changes Current events Random page Help Project talk Request account.

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Skorohod, "The theory of stochastic processes" , 1 , Springer Translated from Russian MR Zbl

Note: You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes from to Many mathematical formulas are broken, and there are likely to be other bugs as well. These will most likely not be fixed. We start with some definitions: 1. Stochastic processes A stochastic process is a sequence of random variables X 0 , X 1 , Interpretation: A random process that evolves over time. The elements of this family are called measurable sets.

Any probability space includes a σ-algebra that describes what events are assigned probabilities in the space. For discrete spaces this is often all subsets of Ω. For continuous spaces a restricted σ-algebra is needed to avoid paradoxes; the usual choice for e. For our purposes the useful property of σ-algebras is that they can be used to represent restricted information.

For example, suppose I roll two six-sided dice but only tell you what the sum of the two dice is. What I see is a probability space with 36 elements all outcomes of the two dice. What you see is a random variable with only 11 values the possible sums You can detect whether some events occured "is the sum even? Filtrations With a stochastic process, it is natural to talk not only about the value of the process at time t which is just the random variable X t , but also what we know at time t.

In practice, we are never going to actually do this. We won't prove these facts here, see e. We can think of this process as describing the state of our finances after we've been playing in a casino for a while.

If the casino is perfectly fair unlike what happens in real life , then each bet we place should have an expected return of 0. But this local property has strong consequences that apply across long intervals of time, as we will see below. Special case: A random ±1 walk is a martingale. So {Y t } is a martingale. What about E[X t ], with no conditioning? In other words, martingales never go anywhere, at least in expectation.

We can apply this to the martingales we've seen so far: For an arbitrary betting strategy on a fair game, I neither make nor lose money on average.

This is probably a bad strategy for ordinary people but it works out well for insurance companies, who charge a small additional premium to make a profit above the zero expected return.

Then we can write X t as , where E[Δ i Δ In other words, Δ i is uncorrelated with Δ Δ i This is not quite as good a condition as independence, but is good enough that martingales often act very much like sums of independent random variables. But in fact we can do much better by using moment generating functions; the result is an inequality known as the Azuma-Hoeffding inequality named for its independent co-discoverers , which is the martingale version of Chernoff bounds.

See Azuma-Hoeffding inequality. Application: the method of bounded differences Basic idea: Azuma-Hoeffding applies to any process that we can model as revealing the inputs x x n to some function f x x n one at a time, provided changing any one input changes f by at most c the Lipschitz condition.

The reason is that when the Lipschitz condition holds, then E[f x x t ] is a martingale that satisfies the requirements of the inequality. This allows us to show, for example, that the number of colors needed to color a random graph is tightly concentrated, by considering a process where x i reveals all the edges between vertex i and smaller vertices a vertex exposure martingale.

In simple terms, T is a stopping time if you know at time t whether you've stopped or not. The first condition says that T is finite with probability 1 i. The second condition puts a bound on how big X T can get, which excludes some bad outcomes where we accept a small probability of a huge loss in order to get a large probability of a small gain. So now we'll prove the full version by considering E[X min T,n ] and showing that, under the conditions of the theorem, it approaches E[X T ] as n goes to infinity.

If we can show that the middle term also vanishes in the limit, we are done. Here we use condition 2. n converges to E[X T ]. So the middle term goes to zero as n goes to infinity. This completes the proof.

Using the full-blown optional stopping theorem is a pain in the neck, because conditions 2 and 3 are often hard to test directly. Let T be the time at which the process stops. We have bounded increments by the definition of the process bounded range also works. Again we have bounded increments but not bounded range! Note that the quantity that is "sub" below or "super" above is always where we are now: so submartingales tend to go up over time while supermartingales tend to go down.

If a process is both a submartingale and a supermartingale, it's a martingale. For supermartingales, the same decomposition works, but now Z t is non-increasing. Azuma-Hoeffding inequality for sub- and super-martingales We can use the existence of the Doob decomposition to prove results about a submartingale or supermartingale X t even if we don't know how to compute it. But then Y t -Y t-1 must lie between -2c t and c t , or we violate the constraints on X t. Note that in each case only one side of the usual Azuma-Hoeffding bound holds.

We can't say much about how fast a submartingale with bounded increments rises or a supermartingale falls , because it could be that Z t accounts for nearly all of X t. Where convex functions turn martingales into submartingales, concave functions turn martingales into supermartingales. This fact is not as useful as one might think for getting bounds: given a martingale, it is almost always better to work with the martingale directly and then apply Jensen's inequality or f afterwards to get the desired bound.

Supermartingales and recurrences When we solve a probabilistic recurrence, we get an upper bound on the work that remains in each state. If we have such a bound, we can get a supermartingale that may allow us to prove concentration bounds on the cost of our algorithm. But since we now have a supermartingale instead of just a recurrence, we may be able to get stronger bounds by using Azuma-Hoeffding. Example: QuickSort For example, let's take QuickSort.

We previously calculated see RandomizedAlgorithms that 2 n ln n is a bound on the number of comparisons needed to sort an array of n elements. Let's try to turn this into a supermartingale.

Recall that QuickSort takes an unsorted array and recursively partitions it into smaller unsorted arrays, terminating when each array has only one element. Let's imagine that we do these partitions in parallel; i. Since each such step reduces the size of the largest unsorted block, we finish after at most n such steps. We expect each block of size n i to take no more than 2n i ln n i comparisons to complete.

Let's consider the partition of a single block of size n i. The cost of the partition is n i -1 comparisons. n-1 we assume that all elements are distinct so that the pivot ends up in a block by itself. The subtracted terms are equal to The -n pays for the n-1 cost on average.

We already knew this. Can we learn anything else? This is a pretty weak bound, although it's much better than we get with just Markov's inequality. There are much better ways to do this. Martingales [ FrontPage ] [ TitleIndex ] [ WordIndex ].

Navigation menu Personal tools Not logged in Talk Contributions Create account Log in. This technique can be contrasted with the anti-martingale system , which involves halving a bet each time there is a trade loss and doubling it each time there is a gain. Let T be the time at which the process stops. The cost of the partition is n i -1 comparisons. Log in. We can apply this to the martingales we've seen so far: For an arbitrary betting strategy on a fair game, I neither make nor lose money on average. It's also important to note that the amount risked on the trade is far higher than the potential gain.

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