From: ross@math.hawaii.NOSPAM.edu (D. Ross) Subject: Re: convergence in probability vs. with probability one? Date: Wed, 03 Nov 1999 06:49:41 GMT Newsgroups: sci.math,sci.stat.math Keywords: different notions of convergence Andrew Jahns wrote: | In a recent probability class we were introduced to various stocastic | convergence modes. Two of those were convergence in probability and | convergence with probability one. Try as I might, I have been unable to | get a grasp of these concepts. You should ask your professor, she will have an easier time explaining these to you at the blackboard or with paper handy in his office than we will using plain text in a newsgroup. Besides, she's paid to do this:-) David's answer, however unsatisfying, is dead right: you need to know these because they will play an important role later in a variety of important results which cannot even be articulated until you understand these concepts. Alas, this is the way mathematics works. That said... Roughly speaking, the difference is this. X_n->X a.s. (i.e., w/prob 1) means that almost every time you run an experiment from which the RV's X_n and X are defined, X_n will converge to X. The weaker X_n->X in probability means that for any fixed degree of tolerance for the difference between X_n and X (this is the epsilon), and any fixed fraction r% you specify, you can find an N big enough that if you run your sequence X_n for n past N you will guarantee that X_n will be within the specified tolerance of X at least r% of the time. Of course, the particular experimental runs that get you within this tolerance might be different for n and n+1. - David R. ============================================================================== From: "Adam Atkinson" Subject: Re: convergence in probability vs. with probability one? Date: 03 Nov 99 19:27:21 +0000 Newsgroups: sci.math,sci.stat.math On 02-Nov-99 14:24:58, Andrew Jahns said: >In a recent probability class we were introduced to various stocastic >convergence modes. Two of those were convergence in probability and >convergence with probability one. OK >1) Why do I care? I don't know >What is the utility of these concepts? They're definitions you need to know about for later use, just like limits. Quite a lot of probability theorems look something like "If you have (one of these) then (this thing calculated from it) converges to (something) in (some manner)". If you can get convergence with probability one then that's very nice. Sometimes the best you can do is something less impressive. Sometimes you can get convergence with probability 1 if your "one of these" satisfies more stringent conditions. >2) Which important sequences converge in one mode and not the other? Well, convergence w.p.1 implies convergence in probability. So we only have to worry about the other way round. The example in "Probability and Random Processes" by Grimmett and Stirzaker of something which converges in probability but NOT w.p.1 is: X_n = 1 with probability 1/n 0 with probability 1 - 1/n It's pretty clear that X_n -> 0 in probability. For any epsilon, Pr(X_n > epsilon) tends to zero as n tends to infinity. However, we do NOT have convergence w.p.1. For that to happen, it would have to be the case that P(X_n -> 0) = 1. i.e. we want "with probability 1, X_n is 1 only finitely many times". But this isn't true. G&S's proof is more or less: The probability that X_n is 0 for all n from m onwards is lim r-> infinity of (1-1/m)(1-1/(m+1))...(1-1/r) = lim M->infinity of (m-1 / m)(m / m+1)(m+1 / m+2)...(M / M+1) = lim M->infinity of (m-1)/(M+1) which is zero for all m. Is this an "important sequence"? No, but snappy memorable counterexamples aren't always important. G&S is good for counterexamples. If you're trying to break things in probability, try checking if they're even true for evil things like the Cauchy distribution, or the Cantor distribution. >3) Is convergence with probability one somehow better than convergence >in probability? Yes. The first implies the second, but not the other way round. -- Adam Atkinson (ghira@mistral.co.uk) "That's the biggest shark I've ever seen" he said, superficially.