Suppose you have a positive sequence $X_1,X_2,\dots$ of i.i.d. random variables with the property that $$ \mathbb{E}[\log(X_1)]<\infty. $$

Is it true that $$ \limsup_{n\to\infty} e^{-n}\sum_{k=1}^n e^k X_k < \infty? $$ If so, does there exists a limit in some sense?

I don't know exactly what is covered in the literature. I would be grateful for any suggestions.


No, it is not. If the $X_i$ are not almost surely bounded, so that for every $N$ there is some positive probability that $X_i > N$, then almost surely there is an infinite increasing sequence $n_N$ such that $X_{n_N} > N$, and $$e^{-n_N} \sum_{k=1}^{n_N} e^{k} X_k \ge X_{n_N} > N$$

  • $\begingroup$ I am not sure how this shows that $\limsup_{n\to\infty} e^{-n}\sum_{k=1}^n e^k X_k =\infty$. As far as I can tell, this argument gives me that $\mathbb{P}(\limsup_{n\to\infty} e^{-n}\sum_{k=1}^n e^k X_k > N)\leq \epsilon$. $\endgroup$ – Bati Apr 19 '16 at 18:03
  • $\begingroup$ Ah sorry I misread your answer. $\endgroup$ – Bati Apr 19 '16 at 18:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.