Thread: Convergence of a nonnegative submartingale

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

Hey alphabeta89.

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

Ya, Can I use X_n = n?

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

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

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

Right?

Well then it looks OK.