Victor Yu. Korolev

This paper shows that an arbitrary scale mixture of normal laws can be a stationary distribution of a stochastic difference equation (first-order autoregressive scheme) with random coefficients.

An example is given of what a (random) diffusion coefficient should look like for a particular mixture to be a stationary distribution.

This study presents a modified vector autoregression (VAR) method for forecasting the quality metrics of overlay channels. The modification involves the introduction of weighted coefficients for time series quantiles, with two distinct approaches proposed for calculating these weights: exponential weighting (EVAR) and linear weighting (LVAR). Experimental results demonstrate that the proposed method improves prediction accuracy by 2.6% to 25.2% compared to classical AR and VAR methods, albeit at the expense of higher computational complexity.

In this paper, the definitions of generalized Student’s distributions are extended to a wider set of parameters of these distributions and multiplication theorems are given that allow the generalized Student’s and Lomax’s distributions to be represented as scale mixtures of the same distributions but with larger parameters. A similar result is obtained for beta distributions. Analogs of multiplication theorems are obtained for the classical Student’s and Lomax’s distributions as corollaries; in particular, it is shown that the Student’s distribution can be represented as a scale mixture of the Student’s distribution with a large number of degrees of freedom. A representation of strictly stable distributions concentrated on the positive semiaxis is also obtained as scale mixtures of a special distribution that is not stable. This alternative representation complements the multiplication theorem for such strictly stable laws.