Let {X n : n = 0, 1,…} be a random walk with state space {0, 1, 2,…} and transition…

Let {Xn : n = 0, 1,…} be a random walk with state space {0, 1, 2,…} and transition probability matrix

where 0 n→∞ pnij exists and is independent of i. Then find the limiting probabilities.

Example 12.12

(One-Dimensional RandomWalks) In a large town, Kennedy Avenue is a long north-south avenue with many intersections. A drunken man is wandering along the avenue and does not really know which way he is going. He is currently at an intersection O somewhere in the middle of the avenue. Suppose that, at the end of each block, he either goes north with probability p, or he goes south with probability 1 − p. A mathematical model for the movements of the drunken man is known as a one-dimensional random walk. Let α be an integer or −∞, β be an integer or +∞, and α n : n = 0, 1,…} with state space {α, α + 1, . . . , β}, a subset of the integers, finite or infinite. Suppose that if, for some n, Xn = i, then Xn+1 = i + 1 with probability p, and Xn+1 = i − 1 with probability 1 − p. That is, one-step transitions are possible only from a state i to its adjacent states i − 1 and i + 1. Clearly {Xn : n = 0, 1,…} is a Markov chain with transition probabilities