Stationary distribution of an infinite Markov chain

Question

The exercise is asking for the stationary distribution, the estimated time to get from state $0$ to state $4,$ and to conclude if the chain is time-reversible.

So I have the following transition matrix:

$$P=\left[\begin{matrix} 1/2&1/2&0&0&0&\cdots\\ 1/2&0&1/2&0&0&\cdots\\ 1/2&0&0&1/2&0&\cdots\\ 1/2&0&0&0&1/2&\cdots\\ 1/2&0&0&0&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots \end{matrix}\right]$$

This means $P_{i,0} =1/2,$ $P_{i,i+1} = 1/2,$ and $P_{ij}=0$ in every other case.

I have progressed up to this point:

\begin{align*} n_1&=\frac12\,n_1+\frac12\,n_2+\frac12\,n_3+\dots\\ n_2&=\frac12\,n_1\\ n_3&=\frac12\,n_2\\ &\vdots\\ n_k&=\frac12\,n_{k-1}\\ \sum_{k=1}^\infty n_k&=1. \end{align*}

I'm not sure how to solve the problem, though.

Any help would be very much appreciated.

Answer 1

We can rewrite manipulate the first equation by noticing that $\pi_1+\pi_2+\pi_3+\cdots=1$. This gives us the following:

$$\begin{align} \pi_1&=\frac12\pi_2+\frac12\pi_3+\frac12\pi_4+\cdots\\ &=\frac12\left(\pi_2+\pi_3+\pi_4+\cdots\right)\\ &=\frac12(1-\pi_1) \end{align}$$

Solving $\pi_1=\frac12(1-\pi_1)$ gives us $\pi_1=\frac13$. Then, each term is half of the previous one ($\pi_n=\frac12\pi_{n-1}$), so we have $\pi_n=\frac13\left(\frac12\right)^{n-1}$.

Hope it was helpful!