Let P be a stochastic matrix with S = {1,. . . , N} which is associated with an irreducible chain. Often, for convergence reasons, we want the Markov chain to be aperiodic. However, in simulation, we want the chain to explore space quickly. For these reasons we modifies the string by introducing a parameter .

Let's pose m = min(pii) for 1iN and a = m/(1-m) .

The new stochastic matrix becomes, (1 - ) P + I, a <1, I is the identity matrix,

Show that the matrix (1 - ) P + I is a stochastic matrix for all [a, 1].