A chain with states $1,2,...,\rho$ has a matrix whose first and last rows are $(q, p, 0,...,0)$ and $(0,...,0, q, p)$. In all other rows $p_{k,k+1} = p, p_{k, k-1} = q$. Find the stationary distribution.
the stationary distribution $\pi$ satisfies $\pi P = \pi$, however, even for $\rho = 5$ I'm having trouble noticing a pattern for the components of $\pi$ that would lead me to find the stationary distribiution. Some hints would be greatly appreciated!
Just some thoughts :
If $\rho < \infty$ then stationary distribution exists . To show this first show non-zero stationary measure must exist , then normalize it .
Method 1: find basis of $ker(I-P)^T$ , then normalize .
Method 2:
Stationary measure $\mu$ has to satisfy
$$
\left\{ \begin{array}{ccc}
\mu_{0}(1-q) = \mu_{1}q\\
\mu_{\rho}(1-p) = \mu_{\rho-1}p\\
\mu_{k} = \mu_{k-1}p + \mu_{k+1}q & \forall k \in [1,\rho-1]\\
\end{array} \right.
$$
When either $p=0$ or $q=0$ , obviously we have $\pi_0=1 , \pi_\rho=1$ respectively .
Otherwise solve the above recurrence relation for $\mu$ , then normalize .
Method 3: (not sure) Stationary distribution may be solved from detailed balance equation similar to here .
Because your chain is similar to birth death chain , and we know stationary distribution exists , also we must have $\pi_{0}p = \pi_{1}q$ .
There're probably more ... but you could first try above methods and see if they actually work .
