Suppose there is a Markov chain with K states {1,2,...,K} arranged clockwise on a circle. $Z_0$ = 1 The probability of moving clockwise is $p$ and the probability of moving counterclockwise is $1-p$. Then what's the probability that the Markov chain complete a clockwise circle before a counterclockwise one? In addition, I'm also interested in the long run average number of loops completed, i.e., $\lim_{n\to\infty} L(n)/n = ?$, where $L(n)$ is the number of loops completed by time n.
I don't know how I should approach this question. Any help is appreciated!
The first question is equivalent to a random walk on $\{-K,-(K-1),\dots,0,\dots,K-1,K\}$ starting at $0$ and taking a step to the right with probability $p$ or to the left with probability $1-p$, and asking the probability that the walk reaches $K$ before $-K$; presumably this is a standard problem that has been answered before.
For the second question, the expected number of (signed) clockwise steps per turn is $p-(1-p) = 1-2p$, and so the expected number of (signed) clockwise loops in $n$ steps is $n(1-2p)/K$. It becomes more difficult if you care about unsigned loops, "starting over" every time a loop is completed.