Q 3 Let X be a Markov chain on 0 1 m with transition matr

Q 3 Let X, be a Markov chain on {0, 1 .. . , m} with transition matrix Show by induction (or otherwise) that

Solution

Let Xn-m/2 = Yn

E(Yn) = =E(Xn) - m/2

Let n =1

E(Y1/x0 =i) = E(y1,x0)/E(X0) = (i-m/2) (1-2/m)¹

Hence true for n =1

Let it be true for n =k

Then E(Yk/x0=i) =  (i-m/2) (1-2/m)^k

E(Y_k+1/x0=i) =  (i-m/2) (1-2/m)^k(1-2/m)

= (i-m/2) (1-2/m)^k+1

Thus proved by induction