mean square convergence vs almost sure convergence I saw a few examples that show that almost sure convergence doesn't imply convergence in mean square. Can anyone find an example of a random series that converges in mean square but doesn't converge almost surely?
Look at 

X[n,w]=exp(-n^2*(w-1/n)) where w is in [0,1] (set of all out comes) and n is the time index.

Only for w=0 it does not converge so it almost surely converges (prob 1). But when it does not converge (i.e. w=0) the sequence goes to infinity exponentially fast making it not converging in mean square sense.
Look at 

X[n,w]=exp(-n^2*(w-1/n)) where w is in [0,1] (set of all out comes) and n is the time index.

Only for w=0 it does not converge so it almost surely converges (prob 1). But when it does not converge (i.e. w=0) the sequence goes to infinity exponentially fast making it not converging in mean square sense.