If they're 0,1 valued, then the following may be what you need. It's call Levy's Borel-Cantelli Lemmas:

Suppose that for natural numbers n, E_n \in F_n (a sigma-algebra). Define Z_n = \sum {1\leq k \leq n} I{E_k}, the number of E_1, ..., E_n which occur. Set e_k = P(E_k | F_{k-1}), and Y_n = \sum_{1\leq k \leq n} e(k). Then, almost surely, (a) Y_\infty < \infty implies Z_\infty < \infty (b) Y_\infty = \infty imples Z_n / Y_n --> 1.

This allows, in essence, for you to use the Borel-Cantelli lemmas even when the variables are dependent, as long as many variables are mostly independent from the earlier ones.

I know it isn't the strong law, but in many circumstances that you'd want a strong law (but don't have it) this suffices.

My reference is "Probability with Martingales", by D. Williams, and this is Theorem 12.15 (with proof) on page 124.

If they're 0,1 valued, then the following may be what you need. It's call Levy's Borel-Cantelli Lemmas:

Suppose that for natural numbers $n$, $E_n \in F_n$ (a sigma-algebra). Define $Z_n = \sum _{1\leq k \leq n} I_{E_k}$, the number of $E_1,\ldots,E_n$ which occur. Set $e_k = P(E_k | F_{k-1})$, and $Y_n = \sum_{1\leq k \leq n} e(k)$. Then, almost surely, (a) $Y_\infty < \infty$ implies $Z_\infty < \infty$ (b) $Y_\infty = \infty$ imples $Z_n / Y_n \rightarrow 1$.

This allows, in essence, for you to use the Borel-Cantelli lemmas even when the variables are dependent, as long as many variables are mostly independent from the earlier ones.

I know it isn't the strong law, but in many circumstances that you'd want a strong law (but don't have it) this suffices.

My reference is "Probability with Martingales", by D. Williams, and this is Theorem 12.15 (with proof) on page 124.