136
This result follows from Theorem B3 by specializing it to nxn stochastic primitive
matrices via Theorem B4, Proposition B and Proposition B2. A very significant conse-
quence is the following.
Proposition B4: If P is an n x n stochastic primitive matrix and x an arbitrary
n-dimensional probability vector, then
lim (x- = qT) -'r = qr
k->~
where q is the unique vector asserted in Proposition B2.
Definition B3: If the n x n stochastic primitive matrix P in Theorem B4 and Propositions
B1-B4 is the state transition matrix of a time homogeneous Markov chain, then the
unique vector q asserted in Proposition B2 is the stationary distribution of the Markov
chain.