Given a random walk for which 0 is an absorbing state and such that from any positive state, the process is equally likely to go up or down one unit, $p_i = q_i = 1/2$. Also note, $R_0 = 1$ (for the absorbing state to be 0).
a) Show that the random walk is a nonnegative martingale. I am unsure how to proceed. Any help would be much appreciated!
To prove that $(R_n)$ it is a martingale you need to show that $E(R_n|R_{n-1},\ldots, R_0) = R_{n-1}.$ Split up cases into when $R_{n-1} = 0$ and when $R_{n-1}\ne 0.$
Let $R_{n-1}\ne 0.$ Since it's equally likely to go up one or down one from there, $R_n$ is (conditionally) equally likely to be $R_{n-1}-1$ or $R_{n-1}+1$. So what's the conditional expected value of $R_n$?
When $R_{n-1}=0,$ then since it's an absorbing state, $R_n=0$ with probability one. So does the conditional expected value of $R_n$ equal the value of $R_{n-1}?$
Then you need to show it's non-negative, but that's pretty clear.