I have a sequence of positive random variables $Xt$, and I am trying to determine the convergence of $$\sum_{t\ge1} \frac{1}{E(T_t)}.$$
I know that $$P(\limsup(Xt=O(t))=1.$$
This implies that $X_t\le ct$ infinitely often, and so $\frac{1}{T_t} \ge \frac{1}{ct}$ i.o. , and I was hoping this would prove that $$\sum_{t\ge1} \frac{1}{E(T_t)} \geq \sum_{t\ge1} \frac{1}{ct}=\infty $$
but since it can fail infinitely often this doesn’t hold. This could be resolved if $X_t < ct$ only finitely many times, however I was able to prove that $$P(\liminf(Xt=O(t))=0,$$ and so $X_t <ct$ infinitely often as well.
Are there any other approaches I can take to see whether the sum will converge or diverge, probabilistically?