Show a random walk is transient

Question

I was going through some problems related to Markov chains and I got stuck on this bit:

We are given a random walk on $Z$, defined by the transition matrix $p_{i,i+1}=p$ and $p_{i,i-1}=1-p$. How to show that if $p\neq 0.5$ the walk is transient?

Answer 1

Since the process is irreducible, we can assume without loss of generality that $X_0=0$, and it suffices to show that $\mathbb P(N_0<\infty)<1$, where $$N_0=\inf\{n>0: X_n=0\}$$ is the time until the first return to $0$. Let $$F(s) = \mathbb E\left[s^{N_0}\right]$$ be the generating function of $N_0$. It can be shown through some computation that $$F(s) = 1 - \sqrt{1-4p(1-p)s^2}.$$ It follows that $$\mathbb P(N_0<\infty) = F(1) = 1 - \sqrt{1-4p(1-p)}<1.$$