Let $(S_n)_{n \geq 0}$ be a random walk on the integers with transition probabilities $$p_{i, i-1} = \left\{\begin{matrix} p, & \text{if $i$ is even} \\ 1 - p & \text{if $i$ is odd}, \end{matrix}\right. \ \ \ \ p_{i,i+1} = \left\{\begin{matrix} 1-p, & \text{if $i$ is even} \\ p & \text{if $i$ is odd}, \end{matrix}\right. \ \ \ \ 0 < p < \frac{1}{2}.$$ I want to find out if this random walk is recurrent or transient. There are a couple ways to show recurrence of random walk that I know of:
- Show that $\sum_{n=0}^{\infty}P(S_{2n}= 0) = \infty$
- Show that $P(T< \infty) = 1$ where $T$ is some hitting time
- Exhibit a stationary distribution
However, I do not know which one to use here, or which one would be the easiest. How does one do this problem? And, if you don't mind, can you explain how one could look at a problem like this and know which "method" to use?
Thanks!
Combine the steps in pairs. The resulting random walk on the sites of a given parity remains in place with probability $p^2+(1-p)^2$ and goes in either direction with equal probability $p(1-p)$. Since the symmetric random walk is known to be recurrent, this one is as well; the probability to remain in place just slows things down.
I’m afraid I don’t know (or at least don’t have reflective access to any knowledge I have about) how one looks at a problem like this and knows which method to use – I think it’s mostly experience.