Definition (Gaussian process)

A Stochastic Process $(X_{t})_{t\in\mathbb{R}}$ is called a Gaussian Process if for any finite collection of points $t_1, t_2, \dots, t_n \in \mathbb{R}$, the Random Vector $\mathbf{X} = [X_{t_{1}}, X_{t_{2}}, \dots, X_{t_{n}}]^\top$ follows a Multivariate Gaussian distribution: $$\mathbf{X} \sim \mathcal{N}(\mathbf{m}, \mathbf{K})$$where:

• $\mathbf{m} = [m(x_1), m(x_2), \dots, m(x_n)]^\top$ is the mean vector.
• $\mathbf{K}$ is the $n \times n$ Covariance matrix with entries $K_{ij} = k(x_i, x_j)$.