In the following series of questions, we will study the stationary distribution of the simple symmetric random walk.

g ;, Egauchospace.ucsb.edu (f, o ' ] ' l ' 3. (10 points) In the following series of questions, we will study the stationary distribution of the simple symmetric random walk. For a Markov chain with transition matrix A, we know that to nd the stationary distribution one needs to solve the system of linear equations ATA = AT. The main di'iculty with simple symmetric random walk is that the state space is the set of all integers. Hence, one Will have to deal with innite dimensional matrix and vector. However, you can overcome the challenge by following my guidance. (1). First consider a 3 X 3 transition matrix A with state space {1,2,3}. Denote by AT = (A1, A2, A3) one stationary distribution. Show that A1, A2, and A3 satisfy A12A1 + (A22 1))\\2 + A32A3 = 07 (1) where Aij means the element in the ith row and jth column of A