Informatics and Applications

2026, Volume 20, Issue 1, pp 30-38

ON NORMAL VARIANCE-MEAN MIXTURES AS STATIONARY DISTRIBUTIONS OF A STOCHASTIC DIFFERENCE EQUATION WITH RANDOM COEFFICIENTS

  • V. Yu. Korolev
  • N. R. Romanyuk

Abstract

It is shown that an arbitrary normal variance-mean mixture can be a stationary distribution of a stochastic difference equation (that is, in the first-order autoregressive scheme) with random coefficients. An example is presented of what the random drift and diffusion coefficients should look like in order that a specified mixture is a stationary distribution. It is demonstrated that one and the same stationary distribution can occur with different forms of the coefficients. In terms of the closeness of coefficients, some estimates are presented for the closeness of the distributions of random autoregressive sequences of the first order. It is also shown that the stationary mode of the first-order autoregressive process with random coefficients possesses the property of stability in the sense that small deviations of the distribution of the initial term of the autoregressive sequence from the stationary distribution corresponding to the given coefficients guarantee small deviations of the distributions of the rest terms of the sequence from this distribution.

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