Visits from a transient random walker on the integers

Question

Consider a random walk $\{S_n\}$ on $\mathbb{Z}$ with forward probability $p>\frac12$. It is known for such a transient RW that each site is a.s. visited only finitely many times. However, is it true that $$P[\text{Each site to the right of the origin is visited less than }k\text{ times}]=0$$ for all $k>1$?

Intuitively, it seems obvious. However, I'm having a tough time proving this rigorously. Any suggestions or hints are welcome.

Answer 1

This is correct. The statement that each site to the right of the origin is visited less than $k$ times is exactly the same as the probability that the random walk will never back track over given fixed state $k$ times. But at any state $n>0$ there is strictly positive probability $p^k(1-p)^k$ that the next $2k$ steps just jump back and forth between $n$ and $n-1$.

Here we are in a position to apply the principle that "whatever always stands a reasonable chance of happening will almost surely happen - sooner rather than later". In particular I am quoting David Williams' (Probability with Martingales)

Lemma 10.11 Suppose $T$ is a stopping time such that for some $N$ and some $\epsilon>0$ we have for every $n$ that $P(T \leq n + N| \mathcal{F}_n)\geq \epsilon$ a.s. implies $E[T]<\infty$. In particular $T$ is finite almost surely.

Applying it to this situation (here we are letting $T$ be the first time any state is hit for the $k$-th time), and $\epsilon=p^k(1-p)^k, N=2k$) we find in fact that with probability 1, some site to the right of the origin is visited at least $k$ times. That is $$ P[(\text{Each site to the right of the origin is visited less than }k\text{ times})^c]=1 $$ which proves your claim.