Consider a coordinated random walk with stay. That is, a walker can move to the right, to the left, or not move at all. Given that the move in the current step is to the right, then in the next step it will move to the right again with probability

a, to the left with probability

b, and remain in the current position with probability 1 − a −

b. Given that the walker did not move in the current step, then in the next step it will move to the right with probability

c, to the left with probability

d, and not move again with probability 1 − c −

d. Finally, given that the move in the current step is to the left, then in the next step it will move to the right with probability g, to the left again with probability h, and remain in the current position with probability 1 − g − h. Let the process be represented by the bivariate process {(Xn, Yn), n = 0, 1, 2,...}, where Xn is the location of the walker after n steps and Yn is the nature of the nth step (i.e., right, left, or no move). Let π1 be the limiting probability that the process is in the “right”

state, π0 the limiting probability that it is in the “no move” state, and π−1 the limiting probability that it is in the “left” state. Let  = [π1, π0, π−1], where

π1 + π0 + π−1 = 1.

a. Find the values of π1, π0, and π−1.

b. Obtain the transition probability matrix of the process and show that the process is a quasi-birth-and-death process.

Random Walk 261