1.10 The gaussian process

A random process $X(t)$ is Gaussian if the joint density of the $N$ amplitudes $\{X(t_1), ,, X(t_N)\}$ comprise a Gaussian random vector. The elements of the required covariance matrix equal the covariance between theappropriate amplitudes: $K_{i, j}=K_X(t_i, t_j)$ . Assuming the mean is known, the entire structure of the Gaussian random process is specified once the correlationfunction or, equivalently, the power spectrum is known. As linear transformations of Gaussian random processes yieldanother Gaussian process, linear operations such as differentiation, integration, linear filtering, sampling, andsummation with other Gaussian processes result in a Gaussian process.