Convergence of a nonnegative submartingale

**alphabeta89** · March 2nd 2013, 07:19 AM

Hi, how to I show that a non-negative submartingale need not converge almost surely?

**chiro** · March 2nd 2013, 09:55 PM

Hey alphabeta89.

Do sub-martingale always increase (or at minimum, remain unchanged)?

**alphabeta89** · March 3rd 2013, 12:05 AM

Ya, Can I use X_n = n?

**chiro** · March 3rd 2013, 12:10 AM

That seems reasonable, but can you please remind what almost surely refers to?

**alphabeta89** · March 3rd 2013, 12:41 AM

If $X_n$ converge almost surely to $X$ , then

$P(\lim_{n\to\infty}X_n = X) = 1$ .

Right?

**chiro** · March 3rd 2013, 02:23 PM

Well then it looks OK.

Thread: Convergence of a nonnegative submartingale