We will now show how the upper bounds for the variational distance
and the second largest absolute
value
of the
eigenvalues
of the transition
matrix
derived in Section 2.3 can be used
in order to determine upper bounds for the distance
occurring in the th step of the MCMC
simulation via the Metropolis algorithm,
if the simulated distribution
satisfies the following
conditions.
Namely we assume
that
for arbitrary
such that
,
and that the states
are ordered
such that
.
We may thus (w.l.o.g.) return to the notation used in
Section 2.3 and identify the states
and the first natural
numbers, i.e.
.
The probabilities
can thus be written in
the following way:
(56)
where
is a monotonically
increasing function,
and is chosen such that for a certain constant
(57)
and
is an (in general unknown)
factor.
Furthermore, the definition of a Metropolis algorithm for the MCMC
simulation of
requires
that the basis and the differences
are known for
all
,
i.e. in particular that the quotients
are known
for all
.
Let the matrix
of the ,,potential'' transitions
be given by
(58)
Let the acceptance probability be defined as in
(53), i.e.
By (56) and (58) the entries
of the transition matrix
for the MCMC simulation are thus be given as
(59)
and for
(60)
Theorem 3.15
The second largest eigenvalue
of the transition matrix
defined by

has the following upper
bound