Question: 5.5. Consider a stochastic process that evolves according to the following laws: If X,, = 0, then X,,+, = 0, whereas if X,, > 0,

5.5. Consider a stochastic process that evolves according to the following laws: If X,, = 0, then X,,+, = 0, whereas if X,, > 0, then

Xn+1 = (X+1 with probability X-1 with probability.

(a) Show that X,, is a nonnegative martingale.

(b) Suppose that X0 = i > 0. Use the maximal inequality to bound Pr{X ? N for some n ? OIXO = i}.
Note: X,, 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 (X,, = 0). This gambler's ruin problem is discussed fully in III, Section 5.3.

Xn+1 = (X+1 with probability X-1 with probability.

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