Problems: 1). A random walk is defined on the integers {0, 1, 2, 3...

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Problems: 1). A random walk is defined on the integers {0, 1, 2, 3, . . .} with the following transition probabilities: p00 = 1 2 , p01 = 1 2 pn,n+1 = pn,n−1 = 1 2 n = 1, 2, . . . Determine whether the walk is transient or persistent. [Hint: relate the walk to the standard symmetric walk on the integers. The standard symmetric walk is persistent, thus f1,0 = 1. By conditioning on the first step of the walk starting at 0, use this to compute f0,0 for the walk on the half-line given above]. 2). Examine the random walk on the two-dimensional lattice of points {(m, n)} with m, n all positive and negative integers, where at each step there is a random jump to one of the four neighbors with probability 1/4 each. Determine if the walk is transient, positive persistent or null persistent. [Hint: use Stirling’s formula for the asymptotics of n!: n! ∼ n n e −n √ 2πn 3). Suppose that X and Y are independent irreducible aperiodic Markov chains with the same state space S and the same transition matrix P. Let Z = (X, Y ) be the chain with state space S × S and transition matrix P  Z1 = (j, l)|Z0 = (i, k)  = P(X1 = j | X0 = i) P(Y1 = l | Y0 = k) Show that Z is also irreducible and aperiodic. [Hint: use the following theorem: “an infinite set of integers which is closed under addition contains all but a finite number of positive multiples of its greatest common divisor”] 4). Consider a Markov chain on the set S = {0, 1, 2, . . .} with transition probabilities pi,i+1 = ai , pi,0 = 1 − ai for i ≥ 0, where {ai |i ≥ 0} is a sequence of constants which satisfy 0 < ai < 1 for all i. Let b0 = 1, bi = a0a1 · · · ai−1 for i ≥ 1. Show that the chain is (a) persistent if and only if bi → 0 as i → ∞ (b) non-null persistent if and only if P i bi < ∞, and write down the stationary distribution if the latter condition holds. 5). Let {Xn} and {Yn} be independent random walks on the integers, with X0 = x and Y0 = y. At each step Xn moves to the right with probability p and to the left with probability 1−p. For Yn the probabilities are q and 1−q, respectively. Find the probability that Xn = Yn for some n ≥ 0.

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