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3. (16 points) Let (Yn)n1 be a sequence of i.i.d. random variables with P[Yn=1]=P[Yn=1]=41
and P[Yn=0]=21. Define Xn:=i=1nYi for all n1 and X0=1. (a) Explain why (Xn)n0 is a Markov
chain and provide the corresponding transition matrix P and transition graph. (b) Determine the
communication classes and their periodicity. (c) Argue that Pi,j(n) converges for n for all i and j
in S, and determine the corresponding limits limnPi,j(n). (d) Find a stationary distribution for this
Markov Chain.

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3. (16 points) Let (Yn)n1 be a sequence of i.i.d. random variables wi.pdf

  • 1. 3. (16 points) Let (Yn)n1 be a sequence of i.i.d. random variables with P[Yn=1]=P[Yn=1]=41 and P[Yn=0]=21. Define Xn:=i=1nYi for all n1 and X0=1. (a) Explain why (Xn)n0 is a Markov chain and provide the corresponding transition matrix P and transition graph. (b) Determine the communication classes and their periodicity. (c) Argue that Pi,j(n) converges for n for all i and j in S, and determine the corresponding limits limnPi,j(n). (d) Find a stationary distribution for this Markov Chain.