True or False (if true explain why, and if false, give a counterexample) If E_iN

Question

True or False (if true explain why, and if false, give a counterexample) If E_iN(j) = infinity for some i notequalto j, then j is recurrent. Let T be a random variable taking values in the nonnegative integers. If for any n. the event {T > n} is determined by X_0, ..., X_n, and then T is stopping time. In a Markov Chain on an infinite state space, the expected number of visits to a state could be infinite, yet the state is transient. Let p be a transition function on a finite state space, and let j be a state satisfying p(i, j) > 0 for all j. Then j is recurrent.

Explanation / Answer

a. Expected time to present in a state which is recurrent is infinity, hence it is True

b.True From the definition of Stopping time

c.False. For a transient state the number of visits cannot be infinite

d.j is recurent if p(j,j)>0. but it is given in p(i,j)>0 which is false

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