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Let $\epsilon_n$ be a Markov switching GARCH process as defined by Haas et al.

I have a covariance function $\mathbb{E}(\epsilon_{n-\tau}^2 \epsilon_n^2)$ which depends only on the powers of 2 elements: matrix M which is a constant matrix, and $\delta=p_{11}+p_{22}-1$ which is the second largest eigenvalue of the matrix P (which is the transition matrix of a Markov chain).

Can someone explain why the autocovariance function decays at an exponential rate of max $(\rho(M),\delta)$ or is there some known relationship between the exponential decay of the autocovariance function and the eigenvalues?

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