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.



Weighted sum of i.i.d. random variables
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 $$e^{-n\_N} \sum\_{k=1}^{n\_N} e^{k} X\_k \ge X\_{n\_N} > N$$ 


