On the Laplace Distribution as a Stationary Distribution for a Stochastic Difference Equation with Random Coefficients

The stochastic difference equation scheme, namely the first-order autoregressive scheme with random coefficients is considered. The conditions on the coefficients of this equation which provide of a nontrivial stationary distribution of the autoregressive process, which is the Laplace distribution (double exponential) are posed. The stability of such a stationary regime is proven: small deviations of the distribution of the starting random variable from the Laplace distribution guarantee even smaller deviations of the distributions of all subsequent members of the sequence from the Laplace distribution. Both one-dimensional and multidimensional cases are considered.