Consider the inhomogeneous time Markov chain (Xn) on Z which at step k has the following rule. If at 0, it transitions to { 1, 1} with probability 1/(2k) respectively, or otherwise stays at 0. If at x 7E 0, it transitions to 2x with probability 1/2 and to 0 with probability 1/2. Show the following. . Suppose that n, m E N, show that almost surely lim lim inf Pr(X,,+,,, = 0| X,) = 1 \"Tl>00 iii>00 Hint: Break this into cases, according to value of X n, and then give lower bounds for the probability in terms of m and n. The memoryless property of the geometric random variable is helpful. - Using the bound above, show that limn.H30 Pr(X,, = 0) = 1. . Show by Better BorelCantelli that for some m E N sufficiently large, ka q 0 for infinitely many k, almost surely (and hence -I(X,, 2) 0))