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### Recursive Construction of the ,,Past''

• Recall that
• in Section 2.1.3 we showed that a stationary Markov chain with transition matrix and stationary initial distribution can be constructed as follows, where
• we started with a sequence of independent and on uniformly distributed random variables and defined

if and only if

for all , i.e.

 (92)

• The random variables were defined by the recursion formula

 (93)

where the function was given by

 (94)

• If the pair is reversible, then the stationary Markov chain constructed in (92)-(94) can be tracked back into the past in the following way.
• First of all we extend the sequence of independent and on uniformly distributed random variables to a sequence of independent and identically random variables that is unbounded in both directions.
• Note that due to the assumed independence of this expansion does not pose any problems as the underlying probability space can be constructed via an appropriate product space, product--algebra, and product measure.
• The random variables are now constructed recursively setting

 (95)

where the function is defined in (94).

Theorem 2.14
• Let be a reversible Markov chain with state space , transition matrix and stationary initial distribution .
• Then the sequence defined by - is
• a stationary Markov chain with transition matrix and the one-dimensional marginal distribution ,
• i.e., for arbitrary , and

The proof of Theorem 2.14 is quite similar to the ones given for Theorems 2.11 and 2.13 and is therefore omitted.

Next: Determining the Rate of Up: Reversibility; Estimates for the Previous: Definition and Examples   Contents
Ursa Pantle 2006-07-20