The Spectral Rrepresentation of a Discrete Version of Blackwell's Markov Chain

We consider a Markov chain that can be termed a discrete version of Blackwell's example from 1958. It is constructed with the aid of a sequence of independent Markov chains with two states. It turns out its stationary distribution π and transition matrix P are in detailed balance. As a result, the transition operator associated with P is self-adjoint in ℓ2(π), the Hilbert space of all square summable sequences with respect to π. All eigenvalues of P are therefore real, and we give explicit formulae for them. Their corresponding eigenvectors form an orthogonal family in ℓ2(π). Consequently, P can be diagonalized, and we find manageable formulae for Pn, where n≥2.