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 $\alpha$.

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

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