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Exercise 2.5. Let Mo, M1, M2 .. . be the symmetric random walk of Exercise 2.4, and define fo = 0 and In = Mi(Mi+1

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Exercise 2.5. Let Mo, M1, M2 .. . be the symmetric random walk of Exercise 2.4, and define fo = 0 and In = Mi(Mi+1 - Mi ), n = 1,2 .... 1=0 56 2 Probability Theory on Coin Toss Space (i) Show that 2 (ii) Let n be an arbitrary nonnegative integer, and let f(i) be an arbitrary function of a variable i. In terms of n and f. define another function g(i) satisfying En[f(In-1)] = g(1,) Note that although the function g(1,,) on the right-hand side of this equa- tion may depend on ". the only raudoin variable that may appear in its argument is In; the random variable An may not appear. You will need to use the formula in part (i). The conclusion of part (ii) is that the process lo, /1. /2. ... is a Markov process

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