Let $\{X_i\}$ be a stationary sequence of random variables, $S_n = \sum_{i=1}^n X_i$, and $\text{Var}(S_n) \leq C$, where $C$ is a constant. Note $\{X_i\}$ need not be independent. I am wondering if there is still a Strong Law for this sequence:$$\frac{S_n}{n} \xrightarrow{a.s.} \alpha\,,$$where $\alpha$ is a constant.
I am looking for a reference for a theorem or hints that would help me prove the above.
I know the Strong Law for i.i.d. sequence of random variables. Since $\{X_i\}$ is stationary, $X_i$ are identically distributed. So I believe the question is can we relax the independence assumption in the classical Strong Law. $\text{Var}(S_n)\leq C$ implies $\text{Var}(X_i) \leq C$ and $\text{Cov}(X_i, X_j)\leq C$ for all $1\leq i < j\leq n$. So, I loosely know that the random variables are weakly correlated but it would be great to know a formal proof. Any suggestions?
