yyy: add to what’s currently sec. 10.2 of Chapter 2, version 9/10/99, but which may get moved to the new Chapter 8.
Where does not vary with the parameter we get a simple expression for .
In the setting of (yyy Chapter 2 Lemma 37), suppose does not depend on . Then
xxx JF: I see this from the series expansion for – what to do about a proof, I delegate to you!
yyy: belongs somewhere in Chapter 3.
Recall from Chapter 2 section 3 (yyy 9/10/99 version) that for a function with , the asymptotic variance rate is
(11.19) |
where . These individual-function variance rates can be compared between chains with the same stationary distribution, under a very strong “coordinatewise ordering” of transition matrices.
Let and be reversible with the same stationary distribution . Suppose . Then for all with .
Proof. Introduce a parameter and write . Write for at . It is enough to show
By (11.19)
By (yyy Lemma 11.4 above) . By setting
we can rewrite the equality above as
Since is symmetric wsith row-sums equal to zero, it is enough to show that is non-negative definite. By hypothesis is symmetric and for . These properties imply that, ordering states arbitrarily, we may write
where is the matrix whose only non-zero entries are . Plainly is non-negative definite, hence so is .