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A.4 Inhomogeneous Markov Chains
Complete specification of the inhomogeneous Markov chain X requires its initial
probability distribution, q0, and the infinite sequence of state transition matrices,
{Pk} k 2 0. The probability distribution of Xk, k 2 1 is given by
k-I
qk = F0 I1 P..
n=0
If the chain is asymptotically independent of q0, then it is said to be ergodic. Two classes
of ergodicity must be distinguished. The following definitions and theorems elaborate.
Definition All: The inhomogeneous Markov chain X is weakly ergodic if
Vi,j, le E, Vm e K
lim (p*(i, 1) P(j, 1))= 0.
Weak ergodicity does not require that either lim pk(i, 1) or lim p,k(j, 1) exist.
k--+ k --
Definition A12: Any scalar function T(-), continuous on the set of nxn stochastic matrices
P and satisfying 0 < T(P) < 1 is called a coefficient of ergodicity. If in addition
T(P) = O