Let $(\omega_1, \omega_2, \ldots)$ be iid in $\{-1, 1\}$ and $X_k = \sum_{i=1}^k \omega_i$ be a simple one-dimensional random walk.

Let $\tau_n = \min \{i\in\mathbb{N}: |X_i|=n\}$ be the first time the random walk is $n$ steps from the origin. What I am interested in is the distribution of this hitting time -- in particular, I want to know how the following quantity grows with $n$:

$$\sum_{k=0}^\infty (\mathbb{P}(\tau_n \geq k))^2$$

It is easy to see that it is bounded above by $n^2$, by rewriting it as $$\mathbb{E}\left[\sum_{k=0}^{\tau_n} \mathbb{P}(\tau_n \geq k)\right] \leq \mathbb{E}[\tau_n]$$

However, this is only barely not good enough for me -- what I really need to know is that this is $o(n^2)$, not just $O(n^2)$. (Or, I suppose, knowing that it isn't $o(n^2)$, though that would be a bit boring.)

Another related, and I assume easier, question is this: How many moments does $\tau_n$ have, and what are they?

This question is motivated by this question, so the objective is essentially to see if something interesting about a random walk can be derived via these Fourier methods.