The distribution of the maximum of the multivariate AR(p) and multivariate MA(p) processes

We give the cumulative distribution function (cdf) of Mn, the (element-wise) maximum of a sequence of n observations from a multivariate AR(p) process. We do the same for a multivariate MA(p) process. Solutions are first given in terms of repeated integrals and then for the case, where the marginal cdf of the observations is absolutely continuous. The cdf of the multivariate maximum Mn is then given as a weighted sum of the nth powers of the eigenvalues of a non-symmetric Fredholm kernel. The weights are given in terms of the left and right eigenfunctions of the kernel.