Let $p,q$ be probabilities of taking $+1$ and $-1$ steps respectively along $y$-axis, while $r,s$ denote the probabilities of taking $+1$ and $-1$ steps respectively along $x$-axis. It is given that $p+q+r+s=1.$
I am trying to show whether the expected number of returns to origin is finite or infinite. Let $N$ be a random variable that counts the number of returns. Hence:
$EN=\sum_{n=0}^{\infty}P(S_{2n}=0|S_0=0)=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{2n}{n}\binom{n}{k}^2(pq)^k(rs)^{n-k}$
Now by Stirling's approximation:
$EN\approx\sum_{n=1}^{\infty}\sum_{k=1}^{n-1}\frac{(2n)^{2n+\frac12}}{(n-k)^{2n-2k+1}(k)^{2k+1}}(pq)^k(rs)^{n-k}+(1-4pq)^{\frac{-1}{2}}+(1-4rs)^{\frac{-1}{2}}$
How do I test for convergence of the above series?
Besides, is there a better approach to this problem, given that I am not really familiar with Markov chain?