For a simple random walk {Xn
: n = 0, ±1, ±2,...}, discussed in Examples 12.12 and 12.22, show that
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.
Example 3.14
(Gambler’s Ruin Problem) Two gamblers play the game of “heads or tails,” in which each time a fair coin lands heads up player A wins $1 from B, and each time it lands tails up, player B wins $1 from A. Suppose that player A initially has a dollars and player B has b dollars. If they continue to play this game successively, what is the probability that (a) A will be ruined; (b) the game goes forever with nobody winning?
Example 12.22
(Random Walks Revisited)
For random walks, introduced in Example 12.12, to find out whether or not the states are recurrent, we need to recall the following concepts and theorems from calculus.