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.


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 $P(\tau_{1}=1)=.5$

  • $\begingroup$ Yes I had doubts about it as well. Any way to prove what you wrote though? $\endgroup$ – user157279 Dec 3 '14 at 15:00
  • $\begingroup$ Which part are you asking about. The probabilities? They follow from how you've defined your SRW. Think of where the Walk can be after the first step. $\endgroup$ – Michael Dec 4 '14 at 13:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.