5.1 Let N be a positive integer. Let 0 < p < 1 and P be the probability measure so that the distribution of coin flips is that of N iid Bernoulli(p) random variables. Let (X1, X2, . . . , XN ) be the adapted process given by Xk := ( 1, if k = H 1, if k = T Define a process (M1, M2, . . . , MN ) by M0 = 0 and Mk = Pk i=1 Xi for k > 0, and define a process (I0, I1, . . . , IN ) by Ik = X k1 j=0 Mj (Mj 1 Mj ) for k > 0. Suppose that p = 1/2 and N 2. Prove that (I0, I1, . . . , IN1) is not a Markov process, but the process ((I0, M0),(I1, M1), . . . ,(IN1, MN1)) is a two-dimensional Markov process. (Definition 2.5.5 on page 49 of Shreve is the definition of a multidimensional Markov process. See problems 2.4, 2.5, and 2.6 in Shreve for more review on this topic.) Please note: I went through a sketch of this argument in class, and the problem here can