Weighted sum of i.i.d. random variables

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

• 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 $\mathbb{P}(\limsup_{n\to\infty} e^{-n}\sum_{k=1}^n e^k X_k > N)\leq \epsilon$. – Bati Apr 19 '16 at 18:03
• Ah sorry I misread your answer. – Bati Apr 19 '16 at 18:08