# Exercise 7.1 (Gamblers ruin). Let (Xt) 120 be the Gambler's chain on state space Ω =...

###### Question:

Exercise 7.1 (Gamblers ruin). Let (Xt) 120 be the Gambler's chain on state space Ω = {0, 1,2, , N} (i) Show that any distribution r-[a,0,0, ,0, bl on 2 is stationary with respect to the gambler?s (ii) Clearly the gambler's chain eventually visits state 0 or N, and stays at that boundary state introduced in Example 1.1. chain. Also show that any stationary distribution of this chain should be of this form. thereafter. This is called absorbtion. Let Ti denote the time until absorbtion starting from state i (112) We are going to compute the 'winning probabilities': qi:-P(XTN) By considering what happens in one step, show that they satisfy the following recursion (113) (iii) Denote ρ (1-p)/ p. Show that (114) Deduce that (115) and that (116) (iv)* Conclude that (117) NIN-I if p 1/2.

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