Its a mcq question. All information is in the question. If Zn is not specified, it just means its a general equation.

Question 1 Let (Zn)new be a time-homogeneous Markov process. Then (Zn)ner is a martingale. YES NO No answer Question 2 Let (Mn)neNo be a stochastic process with Mo = 10 such that both (My) neNo and (M2 )nen, are martingales. Then (MR)nEN, is also a martingale. YES NO No answer Question 3 Let (Zn)neNo be a time-homogeneous, positive recurrent and aperiodic Markov process on a finite state space which has a stationary distribution. Then the stationary distribution is unique. YES NO No answer Question 4 There exists a time-homogeneous Markov process where each state is tran- sient and which possess a limit distribution. YES NO No answer Question 5 Let (Zn)neNo be a time-homogeneous Markov process and let both i, j be positive recurrent states. Moreover, denote for any states k, l Pk,I := P(Zn = 1 for some n 2 1 Zo = k). Then either paj > 0, or p;; > 0, or both paj > 0 and pji > 0. YES NO No