Let $(X_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of binary random variables with $$P[X_i = 1]=P[X_i = -1] = \frac{1}{2}$$ and let $$S_n = \sum_{i=1}^{n} X_i.$$
I'd like to show that $$P[\lim \sup_{n \rightarrow \infty} S_n = \infty] = 1$$ with the means of basic probability theory and the Borel–Cantelli lemma or Kolmogorov's 0-1 law. Could somebody give me a hint?
Best Answer
By the Hewitt-Savage's 0-1 law, the random variable $\limsup_{n\to \infty}S_n$ is almost surely constant (the constant may be $ +\infty$ or $ -\infty$ and is denoted by $c$). Defining $S'_n:=S_{n+1}-X_1$, the sequence $(S'_n)_{n\geqslant 1}$ has the same distribution as the sequence $(S_n)_{n\geqslant 1}$, hence $c=c-X_1$. Since $X_1$ is not degenerated, then $c\in\{+\infty,-\infty\}$.
By symmetry of $X_1$ hence that of $S_n$, we actually have $c= +\infty$.