Question: 2.5.5 Consider a stochastic process that evolves according to the following laws: If Xn D 0, then XnC1 D 0, whereas if Xn > 0,

2.5.5 Consider a stochastic process that evolves according to the following laws: If Xn D 0, then XnC1 D 0, whereas if Xn > 0, then

2211 Xn+1 Xn+1 with probability X-1 with probability

(a) Show that Xn is a nonnegative martingale.

(b) Suppose that X0 D i > 0. Use the maximal inequality to bound PrfXn N for some n 0jX0 D ig:
Note: Xn represents the fortune of a player of a fair game who wagers $1 at each bet and who is forced to quit if all money is lost .Xn D 0/. This gambler’s ruin problem is discussed fully in Chapter 3, Section 3.5.3.

2211 Xn+1 Xn+1 with probability X-1 with probability

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