Does the Strong Law of Large Numbers Always Apply?

[jetoso]

For the Strong Law of Large Numbers, as far as I know it applies when, let say, the random variables {Xn: n=1,2,...} are iid, hence uncorrelated, their second moments have a common bound and they have a finite mean mu.
What else I must consider? Is there anyother consideretion or case when the SLLN does not apply?

 

[mathman]

There are weaker conditions for the strong law to hold.

For example, the Kolmogoroff Criterion does not require the distributions be identical.
It needs sum (V_k/k²) be a convergent series (V=variance).

Alternatively if the distributions are identical, all you need is the mean - the variance may be infinite.