Simple Random Walk and $n$th zero hitting time

Question

I am reading an example in Durrett's book regarding the $n$th time the random walk hits 0.

Consider a simple random walk, $X_i=1$ or $X_i = -1$ with equal probability. Let $S_n = X_1 + \dots + X_n$.

Let $T_n$ be the $n$th time $S_m$ hits 0. Durrett claims that if $\tau = \inf\{n \ge 1: S_n =1\}$ then if $\tau_1,\tau_2,\dots$ are independent with the same distribution as $\tau$ then $\tau_1 + \dots+\tau_n$ has the same distribution as $T_n$.

The proof is not provided but is relegated in the next section. It says that results in the next section implies the above result but I just can't seem to find it there...

Can anyone direct me to any online references perhaps regarding the above? Thanks a lot.

Answer 1

Are you sure it says $ \tau= \text{inf}\{n\geq 1: S_{n} =1\}$? If it was $\tau=\text{inf}\{n>1: S_{n} =0\}$ this would make more sense, i.e. the nth hitting time distributed the same as n "first hitting times"? Might be a typo.

In fact, it is surely false. Consider $P(T_{1}=1)=0$ but $P(\tau_{1}=1)=.5$