Graphics Programs Reference
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Figure A.7: Transition probabilities between states and macrostates.
η i (n)
ν i|I (n) =
k∈A I η k (n)
represents the conditional probability of being in state i A I given that the
process is in macrostate A I . Note that a necessary and su cient condition
for the process { Y n ,n 0 } to be an ergodic DTMC is that
= p 0 IJ (n)
p ij
j∈A J
for all i A I , and for all n. Indeed, in this case the conditional probabilities
( A.56) depend only on the present (macro)state, and the dependence on n
in ( A.57) and ( A.58) disappears.
Condition ( A.59) is the lumpability condition of the DTMC { X n ,n 0 }
with respect to the partition S 0 . The DTMC { Y n ,n 0 } is then called the
lumped Markov chain.
If only the steady-state behavior is of interest, it is possible to write for the
DTMC { X n ,n 0 }
η j =
η i p ij
η j = 1
Summing (A.60a ) over all states j in macrostate A J , we obtain
η 0 J =
η 0 I p 0 IJ
A I ∈S 0
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