Determine the discrete-time mc with transition matrix

Question

Assignment:

1. Consider a dice with six facets. Let X_n denote the facet turned up at the nth throw. Determine whether the process Y_n = X_n + X_n+1 is a Markov chain or not.

2. Determine the asymptotic state probability vector of the discrete-time MC with transition matrix

3. Consider the discrete-time MC with transition matrix

Assuming the initial state probability vector is p(0) = [0.3, 0.4, 0.3], apply the Chapman- Kolmogorov equations to determine the state probability vector for n = 1, 2, 3, 4. Repeat the computation assuming p(0) = [1, 0, 0]. Finally, compute the asymptotic probability vector, if it exists.