Gaussian process

Assume the Axiom of Choice The Axiom of Choice. A real stochastic process $X=(X_t{)}_{t\in I}$ Stochastic processes and their finite-dimensional distributions is a Gaussian process if, for every integer $n\ge 1$, every time list $t_1,\dots ,t_n\in I$, and every $a_1,\dots ,a_n\in \mathbb{R}$, the random variable

has a normal law $N(m,{\sigma }^2)$ for some $m\in \mathbb{R}$ and $\sigma \ge 0$. Variance zero is allowed, so constant linear combinations are included.

Equivalently, for every such time list the evaluation vector $(X_{t_1},\dots ,X_{t_n})$ has a possibly singular multivariate normal law in the sense of Multivariate normal law, including singular covariance. Indeed, that definition says exactly that every scalar projection of the vector is normal. This also covers repeated times, zero coefficients, and singular covariance matrices; no distinct-time convention is needed for this definition.

Choice is declared because the library's scalar and multivariate normal-law interfaces construct their probability laws and independent standard-normal realizations under AC. The equivalence itself is only an unpacking of the finite-dimensional projection definition and makes no further selection.

Sousi, Section 6.1, defines a Gaussian process through normal finite linear combinations. Yoshida, Section 6.1 (printed p. 173), uses the equivalent finite-dimensional multivariate-normal formulation. The present definition retains singular laws explicitly, as required by the library's normal-law interface.