Spectral Characterisation of Covariance Existence in Markov-Switching Affine Recurrences

Abstract

This paper provides a finite-dimensional spectral characterisation for when the stationary covariance matrix of a finite-state Markov-switching affine recurrence exists. For stochastic recurrences of the form \[ Z_{n+1}=A_{\Theta_{n+1}}Z_n+\zeta_{n+1}, \] strict stationarity is typically governed by negativity of the top Lyapunov exponent of the random matrix products. This condition, however, does not ensure that the stationary law has finite variances, covariances, and Pearson correlations. We show that these second-order objects are governed instead by a Markov-switching Kronecker operator. If \[ T_{ji}:=p_{ij}(A_j\otimes A_j), \] then, under a natural Perron-excitation condition on the innovation covariance, \(\rho(T)<1\) is necessary and sufficient for the stationary solution to be square-integrable. When this condition holds, the regime-weighted second moments solve a finite-dimensional linear system, which yields the covariance matrix explicitly. Hence, this paper translates the mean-square spectral condition into an exact covariance-existence criterion for the stationary distribution. Examples illustrate the separation: the Lyapunov exponent may be negative while the second-order spectral radius exceeds one, so strict stationarity can persist even though variances, covariances, and Pearson correlations are not finite objects.