## Covariance of unconditional density

I do have a minor problem with a simple covariance matrix.

Let x be a first order Markov chain of length n defined by the transition matrix P, and let . Find E(y) and Cov(y). Note that we assume to be dense, and defined by a spatial correlation function where h = |i-j| for i,j = 1,.., n.

The mean and variance are easy to find,

where is the stationary distribution of x.

However, I get stuck when calculating the covariances using the total law of covariance. I get something like
.

From intuition I guess each element should be something like (j > i), but I can't prove it