@Alessandro I wanted to understand what "Simple random walk on Z starting at 0 returns to 0 infinitely often with probability 1" meant. You can say something as follows: Model a simple random walk of length $n$ starting at $0$ by an i.i.d. sequence of random variables $(X_1, X_2, \cdots, X_n)$ where $X_i = \pm 1$ uniformly with probability 1/2. Then $\{S_k = \sum_{i = 1}^k X_i\}_{1 \leq k \leq n}$ is a Markov chain that represents the walk. Call $p_n = \Bbb P(S_n = 0)$. Then you can observe that $\sum_n p_n = \infty$ - this somehow models the fact that the random walk returns to $0$ infinit…