Determining the Rate of Convergence under Reversibility

- Let and be a quasi-positive (i.e. an irreducible and aperiodic) transition matrix.
- Let
be reversible, where
is an irreducible
and aperiodic transition matrix.
- In this case the detailed balance condition (85) implies the symmetry of the matrix where .
- As the eigenvalues of coincide with the eigenvalues of we obtain for all ,
- and the right eigenvectors of can be chosen such that all of their components are real,
- that furthermore are also left eigenvectors of and that the rows as well as the lines of the matrix are orthonormal vectors.

- The spectral representation (30) of
yields for every
- By plugging in
and
we obtain
for arbitrary

- If is even or all eigenvalues
are nonnegative, then

- By plugging in
and
we obtain
for arbitrary
- This shows that is the smallest positive number such that the estimate for the rate of convergence considered in (96) holds uniformly for all initial distributions .

**Remarks**-
- Notice that (97) yields the following more precise
specification of the convergence estimate (96).
We have
- Consequently,

- However, the practical benefit of the estimate (98)
can be limited for several reasons:
- The factor in front of in (98) does not depend on the choice of the initial distribution .
- The derivation of the estimate (98) requires the Markov chain to be reversible.
- It can be difficult to determine the eigenvalue if the number of states is large.

- Therefore in Section 2.3.5 we consider an
*alternative convergence estimate*,- which depends on the initial distribution
- and does not require the reversibility of the Markov chain.
- Furthermore, in Section 2.3.7 we will derive an upper bound for the second largest absolute value among the eigenvalues of a reversible transition matrix.

- Notice that (97) yields the following more precise
specification of the convergence estimate (96).
We have