But was afraid to ask until it was asked for me: Why does anyone care about the distinction between convergence in probability and almost sure convergence?

Some answers:

1. "Suppose a person takes a bow and starts shooting arrows at a target. Let Xn be his score in n-th shot. Initially he will be very likely to score zeros, but as the time goes and his archery skill increases, he will become more and more likely to hit the bullseye and score 10 points. After the years of practice the probability that he hit anything but 10 will be getting increasingly smaller and smaller. Thus, the sequence Xn converges in probability to X = 10.Note that Xn does not converge almost surely however. No matter how professional the archer becomes, there will always be a small probability of making an error. Thus the sequence {Xn} will never turn stationary: there will always be non-perfect scores in it, even if they are becoming increasingly less frequent."

Also, almost sure convergence implies convergence in probability.

2. The most useful intuitive understanding I've been taught is that almost sure convergence guarantees that X_n be far from X (ie. further than any epsilon) only a finite number of times. Convergence in probability leaves open the possibility that X_n will be far from X an infinite number of times.

The best example I have to illustrate that is if you take Y_n as a Bernoulli(1/n) random variable. Clearly Y_n converges to 0 in probability, but it doesn't converge almost surely. Y_n will always be 1 for an infinite number of n's. You can see this from the second Borel-Cantelli Lemma.

Of course, I've got no idea if the distinction has any practical relevance for econometrics.

3. Convergence in probability is a form of weak convergence. Your students should understand the difference between convergence and weak convergence -- the difference is huge. If you have a sequence x_n, then weak convergence means that f(x_n) --> L for some f. This does not mean that x_n converges, but only that some attribute converges.

For example, you can ask, given N asset prices, if the sum of these prices converges to 1, does that mean that each individual asset price converges to something? No. Here f is the operation of taking the sum. It could be average, variance, integration against a test function, the infimum of a large set of integrations against test functions, whatever. ... Weak convergence, point-wise convergence, and uniform convergence are different concepts and useful ideas to understand, and they appear over and over again in different forms whatever branch of math you are studying.

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