Exercise 2.5. This exercise considers computer simulations of a Markov chain by using random number generators. Here, we assume that the Markov chain has a nonempty countable state space E. Let U , U2, ..., Un,... be i.i.d. random variables uniformly distributed over [0, 1]. Given a function K(i,u) : Ex [0, 1] + E and in E E, define a random sequence (Xn;n > 0) by Xo = i, and inductively Xn+1 = k(X, Un+1), n> 0. (2.1) (1) Show that (Xn) is a Markov chain taking values in E: for all integers n > 0 and states i1,..., in, int1 E, it holds that P(Xn+1 = in+1|X, = in,.. , X1 = i1, X, = 10) = P(Xn+1 = int1|Xn = in). (2) Express the transition probabilities Pij of (Xn) explicitly in terms of K. Supplementary Exercises: 3, 11, 13 in [1, Chapter 3]; 5, 16, 20 in [1, Chapter 4). Exercise 2.5. This exercise considers computer simulations of a Markov chain by using random number generators. Here, we assume that the Markov chain has a nonempty countable state space E. Let U , U2, ..., Un,... be i.i.d. random variables uniformly distributed over [0, 1]. Given a function K(i,u) : Ex [0, 1] + E and in E E, define a random sequence (Xn;n > 0) by Xo = i, and inductively Xn+1 = k(X, Un+1), n> 0. (2.1) (1) Show that (Xn) is a Markov chain taking values in E: for all integers n > 0 and states i1,..., in, int1 E, it holds that P(Xn+1 = in+1|X, = in,.. , X1 = i1, X, = 10) = P(Xn+1 = int1|Xn = in). (2) Express the transition probabilities Pij of (Xn) explicitly in terms of K. Supplementary Exercises: 3, 11, 13 in [1, Chapter 3]; 5, 16, 20 in [1, Chapter 4)