Theorem A6: The inhomogeneous Markov chain X is weakly ergodic if and only if there
exists a strictly increasing sequence of positive numbers {k,}, 1 e K such that
SI 1-1 Pk,,k.,) =00
I=0
Definition A13: The inhomogeneous Markov chain X is strongly ergodic if there exists a
probability vector q satisfying
Vi,j e E, Vm e K : lim p(i,j) = q(j).
Thus, strong ergodicity implies convergence in distribution. The unique vector q is analo-
gous to its time-homogeneous chain counterpart in Theorem A4.
Theorem A7: The inhomogeneous Markov chain X is strongly ergodic if it is weakly
ergodic and if for each transition matrix, Pk, of X there exists a left eigenvector qk corre-
sponding to eigenvalue 1, qk is a probability vector, and
|qk(i)-qk+i(i)l < o
k=0ie E
Further, ifq = lim q, then q is the unique vector in Definition A13.
k-4-