Fourier Analysis and Stochastic Processes by Pierre Brémaud

Author:Pierre Brémaud
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where is the (up to a multiplicative constant) left-eigenvector and is the (up to a multiplicative constant) right-eigenvector corresponding to the eigenvalue , and where the multiplicative constants are chosen in such a way that . Therefore

(*)

We know that all the eigenvalues are of modulus lesser than or equal to 1. If we suppose, as we now do, that the chain is aperiodic, then, besides the eigenvalue , all eigenvalues have a modulus strictly less than 1. In particular, the covariance function is absolutely summable and there exists a power spectral density which is easily computed using the fact that, for ,

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