Consider a stochastic process $\{X_t \colon t \in T\}$ indexed by a set $T$. We assume for simplicty that $T \in \mathbb{R}^n$. We assume that for any choice of indexes $t_1, \dots, t_n$, the random variables $(X_{t_1}, \dots, X_{t_n})$ are jointly distributed according to a multivariate Gaussian distribution with mean $\mu = (0, \dots, 0)$, for a given covariance matrix $\Sigma$.

Under this assumptions, the stochastic process is completely determined by the 2nd-order statistics. Hence, if we assume fixed mean at $0$, then the stochastic process is fully defined by the covariance matrix. This matrix can be defined in terms of the covariance function $$ k(t_i, t_j) = \mbox{cov}(X_{t_i}, X_{t_j}). $$ It is well-known that functions $k$ as defined above are admissible Kernels. This fact is often used in probabilistic inference, when performing regression or classification.

Several functions can be suitable kernels, but only a few are used in practise, depending on the application. Given the large amount of related literature, can someone provide an up-to-date list of functions commonly used as kernels in this context?


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