Markov chain

Let X1 , . .. . Xn N (H, 62), let Fn (1) = = n 1=1 denote their empirical distribution, and let @,2 denote the cdf of the distribution N (1, 62). Recall that in the Kolmogorov-Smirnov test, we considered the test statistic T, = Vn sup |F, (1) - @,.621 ER In the Kolmogorov-Lilliefors test, we consider the test statistic Tn = Vnsup |F, (1) - Qua IER It is true or false that T, and T' , have the same distribution for for all n E N? (Refer to the slides.) O True False6. ( 6 pts ) Consider the following symmetric matrix: 1 1 A=1 11 1 1 1 Compute the spectral decomposition of A = QAQwhere Q is an orthogonal matrix and A a diagonal matrix.2. (15 pts) Consider a Markov chain { Xn } with state space S = {0, 1, 2} and transition matrix and transition matrix P = O ON/H HN/H O (1) Let the mapping f : S - S satisfy f(0) = 0 and f(2) = 1 and assume that f(1) # f(2). If Yn = f(Xn), then when is { Yn } a Markov chain? Is {Yn } always a Markov chain? In other words, are functions of Markov chains always Markov chains?2. A Markov chain with state space {1, 2, 3} has transition probability matrix 0.6 0.3 0.1 [Pa = 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (1)] What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (0) Consider a general three-state Markov chain with transition matrix P11 P12 P13 1?: 1021 1022 p23 P31 P32 P33 Give an example of a specic set of probabilities pm; for which the Markov chain is not irreducible (there is no single right answer to this, of course l]