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.