I'm solving this exercise in Klenke's book:
Let $X_1,X_2, \dots $ be i.i.d. nonnegative random variables. By virtue of the Borel-Cantelli lemma, show that for every $c \in(0,1)$,
$$\sum_{n=1}^\infty e^{X_n} c^n \begin{cases}
< \infty \textrm{ a.s.} & \textrm{if } \mathbb E[X_1] < \infty; \\
= \infty \textrm{ a.s.} & \textrm{if } \mathbb E[X_1] = \infty
\end{cases}$$
There are different ways to prove the statement using the Borel-Cantelli lemma (here's a thread with different answers: link)
However I wanted to try a different approach. I defined $S_k := \sum_{n=1}^k e^{X_n} c^n $ which given the nonnegativity of its elements converges from below to $S:= \sum_{n=1}^\infty e^{X_n} c^n $ . We can prove using the 0-1 Law that $S=a$ almost surely where $ a \in [-\infty, \infty]$ is a constant. And now applying the monotone convergence theorem and taking the expectations delivers:
$$
a=\mathbb{E}[S]=\sum_{n=1}^\infty \mathbb{E}[e^{X_n}] c^n =\mathbb{E}[e^{X_1}] \sum_{n=1}^\infty c^n
$$
Which means that $a$ is finite iff $\mathbb{E}[e^{X_1}] < \infty$. This however is not equivalent to the statement in the exercise. Does anyone see where the argument fails?
Best Answer
We have to distinguish the following two lemmata, both of which are a consequence of Kolmogorov's 0-1-law:
Lemma 1
Let $(X_{k})_{k \in \mathbb{N}}$ be a sequence of independent random variables and let $S_{n}:=\sum_{k=1}^{n}X_{k}$.
Then $\mathbb{P}(S_{n} \text{ converges}) \in \{0,1\}$.
Lemma 2
Any random variable $Y$ that is measurable with respect to the tail-sigma-field of such a sequence of independent random variables, is a.s constant.
To prove almost sure convergence, we could apply Kolmogorov's Three-Series Theorem, but that in itself is a consequence of Borel-Cantelli - so no shortcut here.
Finally, Kolmogorov's 0-1 law does not allow us to conclude that the limit $S=\lim S_{n}$ is constant if it indeed exists, since $S$ is not measurable with respect to the the terminal sigma-field.