Partially Gaussian Stationary Stochastic Processes in Discrete Time

TL;DR

This paper introduces a simple example of a stationary discrete-time stochastic process that is non-Gaussian overall but has Gaussian marginals for any fixed number of variables, challenging assumptions about Gaussianity.

Contribution

It provides the first explicit construction of a stationary process with Gaussian marginals yet non-Gaussian joint distributions, highlighting limitations of marginal-based characterizations.

Findings

Existence of non-Gaussian stationary processes with Gaussian marginals

Explicit elementary example for any fixed k

Challenges assumptions about Gaussianity from marginals

Abstract

We present here an elementary example, for every fixed positive integer $k,$ of a strictly stationary nongaussian stochastic process in discrete time, all of whose $k$-marginals are gaussian.