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Alternative Estimate for the Rate of Convergence;
Contrast
Based on the multiplicative reversible version
of the ergodic (but not necessarily
reversible) transition matrix
we will now deduce an
alternative estimate for the rate of convergence
for
; see
Theorem 2.16.
The following abbreviations and lemmata will turn out to be useful
in the proof of Theorem 2.16.
 Proof

 Introducing the notation
we obtain that
and
where the last but one equality follows from the definition
(99) of the matrix
.
 This implies
and 
(107) 
 On the other hand
and thus
as
is a stochastic matrix such that
and therefore
and
 Taking into account (107) this shows the validity of
(106).
We introduce the following notions.
The distance
between
and
can be estimated via the contrast
of
with respect to
as follows.
 Proof

 Taking into account that
, an application
of the CauchySchwarz inequality yields
 This implies the assertion of the lemma.
The rate of convergence
for
can now be estimated based on
 the second largest eigenvalue
of the
multiplicative reversible version
of
the (ergodic) transition matrix
 and the contrast
of the initial
distribution
with respect to the stationary limit
distribution
.
Theorem 2.16
For any initial distribution
and for all
,

(112) 
 Proof

 Let
where
.
 Then for all
and thus
 Moreover, by definition (110) of the
contrast
of
with respect to
we obtain
i.e.,

(113) 
 Now the identity (106) derived in
Lemma 2.6 yields

(114) 
 On the other hand the spectral representation (101)
of
derived in Theorem 2.15 implies
as
,
and
and therefore
 Summarizing our results we have seen that
Next: DirichletForms and RayleighTheorem
Up: Reversibility; Estimates for the
Previous: Multiplicative Reversible Version of
Contents
Ursa Pantle
20060720