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Consider this definition of a Gaussian Process:

For any set $𝑆$, a Gaussian Process ($\mathcal{GP}$) on $𝑆$ is a set of random variables $\{𝑍_𝑑:𝑑\in 𝑆\}$ such that $\forall𝑛 \in \mathbb{N}$, $\forall t_1,...,𝑑_𝑛 \in S$, $\{𝑍_{𝑑_1},...,𝑍_{𝑑_𝑛}\}$ has a multivariate Gaussian distribution.

What if each random variable $Z_t$ itself has a multivariate Gaussian distribution? Can we easily redefine a Gaussian process to say that "[..] the set $\{Z_{t_1}, ..., Z_{t_n}\}$ has a matrix Gaussian distribution"? Do we run into problems if we use this definition?

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I found a definition in Sparse matrix-variate Gaussian process blockmodels for network modeling (Feng Yan, Zenglin Xu and Yuan Qi, 2012):

A matrix-variate Gaussian process is a stochastic process whose projection on any finite locations $U = [u^⊀_1, . . . , u^⊀_n]^⊀$ follows a matrix-variate normal distribution. Specically, given $U$, the zero mean matrix-variate Gaussian process on $M$ has the form: $p(M|U) =\mathcal{GP}_{n,n}(M; 0,K,G)= (2\pi)^{βˆ’n^2/2}\text{det}(K)^{βˆ’n/2} \text{det}(G)^{βˆ’n/2} \text{exp}\{βˆ’\frac{1}{2}\text{tr}(K^{βˆ’1}MG^{βˆ’1}M^⊀)\}$ where $k_{ij}=k(u_i,u_j)$ and $g_{ij}=g(u_i,u_j)$.

No one seems to talk much about matrix-variate Gaussian processes though.

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