Suppose {$X_n\}_{n \ge 1}$ are i.i.d random variables with $E[X_i]=-1$. Let $S_n=X_1+\cdots+X_n$ is the sum of these r.v. and denote $T$ to be the total number of $n$ satisfying $S_n \ge 0$. The problem is to compute $P(T = \infty)$.
I tried some simple distributions and guess that the answer is $0$. Then as $P(T = \infty)=P(S_n \ge 0 \quad i.o.)$, I want to use Borel-cantelli lemma to conclude but the objective is to prove $\sum_{n \ge 1}P(S_n \ge 0) < \infty$, which I cannot prove.
My attempt was to use law of large numbers or chebyshev inequality to estimate $P(S_n \ge 0)=P(|\frac{S_n}{n}+1|\ge1)$ but they do not give a good supereior. Any answer or hint for the problem? Or does the probability $P(T = \infty)$ is not always $0$? Note that the information about variance is not given, so I think there are not many tools we can use.
Best Answer
By the strong law of large numbers, we have
$$\frac{S_n}{n} \to \mathbb{E}(X_1) = -1 \quad \text{a.s}$$
This implies, in particular,
$$S_n = n \frac{S_n}{n} \xrightarrow[]{n \to \infty} - \infty \quad \text{a.s.}$$
and so
$$\{n \in \mathbb{N}; S_n(\omega) \geq 0\}$$
is a finite set for almost every $\omega \in \Omega$. Hence, $\mathbb{P}(T=\infty)=0$.