Law of iterated logarithm for stationary sequences

Let $$(X_i)_{i\in\mathbb Z}$$ be a sequence of random variables which is stationary and ergodic, but not necessarily i.i.d. Does there exist a real function $$f$$ such that the following holds? $$\limsup_{n\to\infty} \frac{X_1+\cdots+X_n}{f(n)}=1,\quad \text{a.s.}$$ This question is related to the general law of iterated logarithm. Note that $$f$$ might depend on the sequence of random variables. Does there exist any existence result in the non-i.i.d. case?

I need the case where $$X_1\geq 0$$ a.s., but $$\mathbb E[X_1]=\infty$$. You can also assume polynomial tail; i.e., $$\mathbb P[X_1>r]\leq Cr^{-\alpha}$$ for some $$C$$ and $$\alpha$$.

• I think you want to say "does there exist a real function $f$ ..." before "Let $(X_i)_{i \in \mathbb{Z}}$ be ..." – mathworker21 Nov 4 '18 at 10:19
• @mathworker21 No, $f$ might depend on the sequence. I updated the text to be more clear. – Ali Khezeli Nov 4 '18 at 10:24
• Something like this? – user10354138 Nov 4 '18 at 10:41
• @user10354138 It still assumes finite variance. – Ali Khezeli Nov 4 '18 at 10:47