Normal Variance-Mean Mixtures as Stationary Distributions of Stochastic Difference Equations with Random Coefficients

It is shown that an arbitrary normal variance-mean mixture can be a stationary distribution of a stochastic difference equation or first order autoregressive process with random coefficients. An example is presented of what the (random) drift and diffusion coefficients must look like to provide that a certain variance-mean mixture is a stationary distribution of such a process. It is also shown that a stationary mode of the first-order autoregressive process possesses the stability property.