# Does the specific sequence of random variables converge almost surely to a given constant?

Suppose, $$\{X_n\}_{n = 1}^{\infty}$$ is a sequence of i.i.d random variables, such that $$P(X_i = 1) = P(X_i = -1) = \frac{1}{2}$$. Now, suppose $$\{S_n\}_{n = 1}^\infty$$ is a sequence of random variables defined in the following way: $$S_n = \Sigma_{i = 1}^n X_i$$. Now suppose $$T_n$$ is the number of integers $$m$$, such that $$\exists k < n, S_k = m$$ and $$\forall k > n S_k \neq m$$ ($$\{T_n\}_{n = 1}^\infty$$ is also a sequence of random variables.

Is it true, that $$P(\lim_{n \to \infty} \frac{T_n}{n} = P(\forall k \in \mathbb{N}, S_k \neq 0))=1$$?

I see, that $$P(\forall k \in \mathbb{N}, S_k \neq 0) = 2P(\forall k \in \mathbb{N}, S_k > 0)$$, but that does not help much.

Also, I used to think, that $$P(\forall k \in \mathbb{N}, S_k > 0) = 0$$, but now I am not so sure about it…