# Question on circular random walk [closed]

A truck transports goods among $$10$$ points located on a circular route. These goods are carried only from one point to the next with probability $$p$$, or to the preceding point with probability $$q=1-p$$.

1. Write the transition probability matrix.
2. Find the limiting stationary distribution.
The transition matrix is the circulant matrix $$M = q \cdot P + p \cdot P^T$$, where $$P$$ is the permutation matrix in the link. Computing the stationary distribution can be done by computing the solution to the system $$(M - I)x = 0$$.
However, rather than solving this system of equations, we can more easily prove that your guess of the stationary distribution $$\pi = (1/10,\dots,1/10)$$ is correct by verifying that $$\pi M = M$$. To see that this holds, note that $$\pi = \frac 1{10} (1,\dots,1)$$, and that the entries of $$(1,\dots,1)M$$ are the column-sums of $$M$$. The only non-zero entries of a given column of $$M$$ are $$p$$ and $$q$$, which means that every entry of $$(1,\dots,1)M$$ will be $$p+q = 1$$, which means that we have $$(1,\dots,1)M = (1,\dots,1)M \implies \pi M = \pi.$$ So, $$\pi$$ is indeed the stationary distribution.