MATH 419 Midterm: MATH 419 2011 Winter Test 2
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0 (a) find the communicating classes and classify each as transient, positive recurrent or null recurrent. You need not justify your answer. (b) find the period of each state. Justify your answers. (c) show the restriction of p to the state space {3, 4} describes an irreducible aperiodic. If true, give a proof; if false, provide a counter-example. If (mn : n z+) is a non-negative martingale satisfying m0 = 0 and e(hmi ) < , then mn converges a. s. and in l1 to an integrable limit. Show that e(x|g) satis es the same de ning properties (in (a) above) as the conditional expectation of an integrable r. v. X. (iii) give an example of an x and g as above where e(x|g) < a. s. but e(x) : (30 points) theory). martingale. 2 (a) assume {zn : n n} are non-negative iid random variables with mean one, fn = i=1 zi (so m0 = 1).
