Proof of almost sure convergence of sum of iid random variables

Question

I tried to prove the following lemma:
Let $X_1,X_2,\ldots$ iid nonnegative random variables with $E[X_1]=\infty$ and let $a\in(0,1)$, thus follows $\sum\limits_{n=1}^\infty a^n exp(X_n)=\infty$ almost sure.

I tried using Borel Cantelli but failed at proving that for $L\in N$ we have $\sum\limits_{k=1}^\infty P(\sum_{n=1}^k a^k exp(X_n)<L)<\infty$.

Has anybody an idea?

Answer 1

Note that $$ \sum_{n=1}^\infty P(a^n \exp(X_n)>1)=\sum_{n=1}^\infty P(X_1/\log(1/a)>n)=\infty $$ by using the fact that $\int_0^\infty P(X_1>x)\, dx=EX_1=\infty$. Borel Cantelli implies that $a^n\exp(X_n)>1$ i.o a.s. Hence $\sum a^n\exp(X_n)=\infty$ a.s.