Consistency and finiteness together imply that the normalized Matrn measures define a real-valued process (X_{1}, X_{2}, ldots)

Consistency and finiteness together imply that the normalized Matérn measures define a real-valued process \(X_{1}, X_{2}, \ldots\) in which \(M_{n} / \Gamma(v)\) is the joint distribution of the finite sequence \(X[n]=\left(X_{1}, \ldots, X_{n}ight)\). This process-a special case of the Gosset process-is not only exchangeable but also orthogonally invariant for every \(n\). Show that the conditional distribution of \(X_{n+1}\) given \(X[n]\) is Student \(t\), with a certain location parameter, scale parameter and degrees of freedom. To what extent is finiteness needed in the construction of the process?