In Probability Theory by A. Klenke (3rd version), in the proof of Theorem 19.35 it is required to show that almost surely:
$$ R_W^+:=\sum_{n=0}^{\infty}\exp\big(\sum_{k=0}^{n}X_k\big)=\infty \\ R_W^-:=\sum_{n=0}^{\infty}\exp\big(-\sum_{k=-n}^{1}X_k\big)=\infty $$ where $X_k$, $k\in\mathcal{Z}$, are i.i.d. random variables with zero expectation and finite variance.
The given explanation is to use the Central Limit Theorem to show that $$ \limsup_{n\to\infty}\sum_{k=0}^{n}X_k>-\infty \quad(1) \\ \limsup_{n\to\infty}-\sum_{k=-n}^{1}X_k>-\infty \quad(2) $$
From this answer (How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?) I am able to show that (1) holds.
My questions mainly related to the strategy of the proof:
- Are we looking at (1) and (2) because they are related to the necessary condition of convergence of the series, that is the $n$-th term vanishes to zero?
- Why the author uses the $\limsup$ and not $\liminf$? I usually struggle to understand when to use one or the other in proofs.
- How can I show that (2) holds?
Let me know if more context is needed. Thanks for your help and suggestions.