Talk:Econometric Theory/Asymptotic Convergence

So far just tried to think of a basic structure to the chapter.--Alistair 23:43, 29 July 2005 (UTC)

I don't know how to use the AMS math packages to get the \xrightarrow{p} symbol for convergence in probability, as such the formatting is slightly off... Also, as is obvious, my formatting of the definition of Convergence in probability is rather clumsy.--Alistair 12:23, 30 July 2005 (UTC)

Is the example of almost-sure convergence correct?
The example is not right. See http://books.google.de/books?id=irKSXZ7kKFgC&pg=PA93&lpg=PA92&dq#v=onepage&q=&f=false.

I’m hoping the following can give a little intuitive understanding of why the example is wrong, and it may help clarify the difference between convergences almost surely and convergence in probability. I apologize if I’m not 100% technically correct.

For fn(x) to converge to η almost surely, the probability that fn(x) = θ has to be zero for all fm(x) for m>n. In the example, the probability of f(x) = θ equals 1/n; therefore, for any n, the probability sequence of fm(x) = θ, for m > n, starting with n, is 1/(n+1), 1/(n+2), 1/(n+3), etc. We need the probability of the union of these events.

The harmonic series, though, diverges. No matter where you start, that is, for any n in mathematics lingo, the sum of the subsequent terms goes to infinity. I hope it is intuitive that if the sum of probabilities is infinite, the union of these probabilities is 1 – this logic can be seen up by referring to the Borel-Cantelli theorem. So not only is the probability that fm(x) = θ not equal to 0, it is equal to 1!

For me, the easy way to remember the difference between convergence in probability and convergence almost surely, is that for convergence in probability, we simply need that the probability approaches the limit as n gets large. But to get convergence almost surely, it has to equal the limit for an infinite number of trials beyond n! Almost sure convergence is similar to having the sequence of probabilites converge in a classical calculus convergence sense. ```` Can someone remove this example? It is plain wrong. Why would P(\lim_{n\rightarrow} X_n \in A) <=\lim_{n\rightarrow} P( X_n \in A) hold in general? It does not. And "proof" uses this. Also, the definition of X_n here and earlier is incomplete, or, if you want syntactically wrong. --Szepi (discuss • contribs) 01:26, 26 January 2011 (UTC)

Example of Convergence in distribution
The example was adapted from Michael Hardy's text in the wikipedia: w:Talk:Convergence of random variables. Albmont (talk) 17:19, 31 October 2008 (UTC)