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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.

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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 $$e^{-n_N} \sum_{k=1}^{n_N} e^{k} X_k \ge X_{n_N} > N$$

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  • $\begingroup$ 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, 2016 at 18:03
  • $\begingroup$ Ah sorry I misread your answer. $\endgroup$
    – Bati
    Apr 19, 2016 at 18:08

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