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Unformatted text preview: Multiplying this times the probability for each (n1 , . . . , ni )
gives the result. 4.3 Limiting Behavior Having worked hard to develop the convergence theory for discrete time chains,
the results for the continuous time case follow easily. In fact the study of
the limiting behavior of continuous time Markov chains is simpler than the
theory for discrete time chains, since the randomness of the exponential holding
times implies that we don’t have to worry about aperiodicity. We begin by
generalizing some of the previous deﬁnitions
The Markov chain Xt is irreducible, if for any two states i and j it is
possible to get from i to j in a ﬁnite number of jumps. To be precise, there is a
sequence of states k0 = i, k1 , . . . kn = j so that q (km 1 , km ) > 0 for 1 m n.
Lemma 4.2. If Xt is irreducible and t > 0 then pt (i, j ) > 0.
Proof. Since ps (i, j ) exp( j s) > 0 and pt+s (i, j )
to show that this holds for small t. Since
lim ph (km h!0 1 , km )/h = q (km pt (i, j )ps (j, j ) it su ces 1 , km ) it follows that if h is small enough we have p...
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