Problem on the markov chain

Question

Solve the following problem:

1. Consider the Markov chain defined by X^(t+1) = _QX^(t)+ E_t,where

E_t∼N(0,1). Simulating X⁽⁰⁾ ∼ N(0,1), plot the histogram of a sample of X^(t) for t≤ 10⁴ and Q = .9. Check the potential fit of the stationary distribution N(0,1/(1 - Q²)).

2. Show that the random walk has no stationary distribution. Give the distribution of X^(t) for t = 10⁴ and t = 10⁶ when X⁽⁰⁾ = 0, and deduce that X^(t) has no limiting distribution.