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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.

• Let denote the family of all functions
• defined on and mapping into the real line
• and let be an arbitrary positive probability function from , i.e. for all and .
• For arbitrary vectors and we denote by the inner product

 (103)

and by the induced norm, i.e.,

• The terms ( -weighted) mean and variance of will be used to denote the quantities

 (104)

and

 (105)

respectively.

Lemma 2.6   For all , it holds that

 (106)

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.

• Let , let and be arbitrary probability distributions on , and let

 (108)

i.e., the distance between and is expressed via the total variation

 (109)

of the ,,signed measure'' .
• If for all we also consider the term

 (110)

which is called the -contrast of with respect to .

The distance between and can be estimated via the -contrast of with respect to as follows.

Lemma 2.7   If for all , then

 (111)

Proof

• Taking into account that , an application of the Cauchy-Schwarz 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

• As the eigenvectors von defined in (100) are a basis of there is a (uniquely determined) vector , such that .
• Moreover, in (102) we have shown that . As we can conclude .
• Furthermore, as and for all we obtain

• Summarizing our results we have seen that

• Because of (113) and (114) this implies

and

• Thus, we have shown that for all and, consequently, the assertion follows from Lemma 2.7.

Next: Dirichlet-Forms and Rayleigh-Theorem Up: Reversibility; Estimates for the Previous: Multiplicative Reversible Version of   Contents
Ursa Pantle 2006-07-20