Systems and Means of Informatics

2026, Volume 36, Issue 1, pp 81-90

ON SCALAR COVARIANCE-MEAN NORMAL MIXTURES AS STATIONARY DISTRIBUTIONS FOR MULTIVARIATE STOCHASTIC DIFFERENCE EQUATIONS

  • V. Yu. Korolev
  • N. R. Romanyuk

Abstract

The paper addresses the problem of describing the stationary distribution for a multivariate stochastic difference equation, specifically, a first-order multivariate autoregressive scheme with random coefficients. It is demonstrated that any scalar covariance-mean mixture of multivariate normal distributions can serve as a stationary distribution within this framework. Such mixtures are characterized by a scalar mixing parameter that simultaneously scales both the vector of expectations and the covariance matrix, thereby establishing an affine dependence between them. These mixtures have proven effective for modeling statistical regularities observed in various disciplines. It is proved that for any scalar covariance-mean mixture of multivariate normal distributions, it is possible to define a stochastic difference equation with appropriate coefficients for which the given mixture is a stationary distribution. The correspondence between the resulting mixture and the behavior of the coefficients that generate the corresponding stationary distribution is discussed. Also, a problem is considered that is in some sense inverse to that was mentioned above: does there exist a stationary distribution for a stochastic difference equation with given coefficients and if yes, then what does it look like. A version of sufficient conditions for the existence of such a stationary distribution for the stochastic difference equation is presented.

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