# Convergence of a nonnegative submartingale

• Mar 2nd 2013, 07:19 AM
alphabeta89
Convergence of a nonnegative submartingale
Hi, how to I show that a non-negative submartingale need not converge almost surely?
• Mar 2nd 2013, 09:55 PM
chiro
Re: Convergence of a nonnegative submartingale
Hey alphabeta89.

Do sub-martingale always increase (or at minimum, remain unchanged)?
• Mar 3rd 2013, 12:05 AM
alphabeta89
Re: Convergence of a nonnegative submartingale
Ya, Can I use X_n = n?
• Mar 3rd 2013, 12:10 AM
chiro
Re: Convergence of a nonnegative submartingale
That seems reasonable, but can you please remind what almost surely refers to?
• Mar 3rd 2013, 12:41 AM
alphabeta89
Re: Convergence of a nonnegative submartingale
If $X_n$ converge almost surely to $X$, then

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

Right?:D
• Mar 3rd 2013, 02:23 PM
chiro
Re: Convergence of a nonnegative submartingale
Well then it looks OK.