Strong Law of Large Numbers for weakly dependent random variables

This question sounds like an exercise: Split the sequence into I sequences of iid random variables. Apply the classical SLLN to each sequence. Recombine.

Tom: Of course it is true with exponential decay of the corellation function, but it is not easy.

The essential difficulty is that one wants to reduce the SLLN to an exponentially growing subsequence of N's. In the classical case, this is done by a martingale inequality. (Prob of a supremum of a martingale is dominated by the probability at end of the martingale.)

Once one moves away from an implicit martingale structure, then tricks have to be employed---of which the most obvious is that if the sequence of random variables is bounded, then obviously you can reduce to an exponentially growing subsequence. This point is much of the content of the paper of Lyons cited already.

Not sure that this would appear in a text book however. My sense is that these considerations are well-known.

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.