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Estimates for the Rate of Convergence; Perron-Frobenius-Theorem

Example
$ \;$ (Weather Forecast)


In general geometric estimates of the form (44) for the rate of convergence can be derived by means of the following so-called Perron-Frobenius theorem for quasi-positive matrices.

Theorem 2.6    

A proof of Theorem 2.6 can be found in Chapter 1 of E. Seneta (1981) Non-Negative Matrices and Markov Chains, Springer, New York.

Corollary 2.3   $ \;$ Let $ {\mathbf{P}}$ be a quasi-positive transition matrix. Then

Proof
 

Corollary 2.3 yields the following geometric convergence estimate.

Corollary 2.4   $ \;$ Let $ {\mathbf{P}}$ be a quasi-positive transition matrix such that all eigenvalues $ \theta_1,\ldots,\theta_\ell$ of $ {\mathbf{P}}$ are pairwise distinct. Then

$\displaystyle \sup\limits_{j\in E}\vert\alpha_{nj}-\pi_j\vert= O(\vert\theta_2\vert^n)\,.$ (45)

Proof
 


Example
$ \;$ (Reaching a Consensus)
see C. Hesse (2003) Angewandte Wahrscheinlichkeitstheorie. Vieweg, Braunschweig, p. 349


Remarks
 



next up previous contents
Next: Irreducible and Aperiodic Markov Up: Ergodicity and Stationarity Previous: Basic Definitions and Quasi-positive   Contents
Ursa Pantle 2006-07-20