= Gaussian marginals do not imply a Gaussian random field
{c}
Let $X$ be a standard <Gaussian random variable> and $S_j$ <independent> symmetric signs, <independent> of $X$. The stationary sequence $W_j=S_jX$ has standard <Gaussian distribution>[Gaussian] marginals and identity <covariance>, but is not jointly <Gaussian distribution>[Gaussian]. For two coordinates its <moment-generating function> is $(1+e^{2t^2})/2$, rather than $e^{t^2}$. Consequently a field's <Gaussian distribution>[Gaussian] one-point densities and two-point <covariance> alone do not justify an <exponential moment of a Gaussian linear functional>.
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