Let $X=(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. $\mathbb{Z}$-valued random variables satisfying the following conditions:
a) For all $n \in \mathbb{N}$ and $k \in \mathbb{Z}$, we have $\mathbb{P}(X_n=k)=\mathbb{P}(-X_n=k)$ (symmetry)
b) There exists $R \in \mathbb{N}$ such that for all $n \in \mathbb{N}$, we have $\mathbb{P}(|X_n| >R)=0$ (finite range)
c) For all $n \in \mathbb{N}$, we have $\mathbb{P}(|X_n| \geq 1)=1$ (movement)
Let $n \in \mathbb{N_0}$ and define $S_n=\sum_{k=1}^{n}X_k$ so in particular $S_0=0$. For $L \in \mathbb{N}$ we then define the stopping time $\tau_{L}=inf\{n \in \mathbb{N}: |S_n|>L\}$
1) Show that $\tau_L$ is almost surely finite, that is $\mathbb{P}(\tau_L < \infty)=1$
Hint: consider the sets $A_n=\{X_k \geq 1 $ for $ k \in \{2Ln+n,\dots,2L(n+1)+n \} \}$ and use Borel Cantelli.
2) Show that there exists a constant $C=C(R)$ such that for large $L \in\mathbb{N}$ we have $\mathbb{E}(\tau_L) \leq C(R)L^2$
I would appreciate any kind of help/solution you can give me with this exercise. Thank you in advance.
1) The sets $\{2Ln+n,\dots,2L(n+1)+n \},n\in\mathbb{N},$ partition $\mathbb{N}$ into blocks of length $2L$, and so the $A_n$ are independent events. Moreover each $A_n$ has probability $1/2^{2L}$ by (a) and (c). So borel-cantelli 2 implies almost surely $A_n$ occurs infinitely often. Since each $A_n$ implies $\{\tau_L<\infty\}$, $$ A_n\subset\{\tau_L\le 2L(n+1)+n\}\subset\{\tau_L<\infty\}, $$ this is almost overkill to show $\{\tau_L<\infty\}$.
2) This follows directly from Wald's second identity, is that available to you?