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Recursive Representation
- In this section we will show
- how Markov chains can be constructed from sequences of independent
and identically distributed random variables,
- that the recursive formulaź (9),
(10), (11) and (13)
are special cases of a general principle for the construction of
Markov chains,
- that vice versa every Markov chain can be considered as solution of
a recursive stochastic equation.
- As usual let
be a finite (or countably
infinite) set.
- Proof
-
- Remarks
-
Our next step will be to show that vice versa, every Markov chain
can be regarded as the solution of a recursive stochastic equation.
- Remarks
-
- If (16) holds for two sequences
and
of random variables, these sequences
are called stochastically equivalent.
- The construction principle (17)-(19)
can be exploited for the Monte-Carlo simulation of Markov chains
with given initial distribution and transition matrix.
- Markov chains on a countably infinite state space can be
constructed and simulated in the same way. However, in this case
(17)-(19) need to be modified by
considering vectors
and matrices
of infinite
dimensions.
Next: The Matrix of the
Up: Specification of the Model
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Ursa Pantle
2006-07-20