The joint distribution of the maximum and minimum of an AR(1) process

Consider a sequence of nn observations from an autoregressive process of order 11 with maximum MnMn and minimum mnmn. We give their joint cumulative distribution function first in terms of nn repeated integrals and then, for the case, where the marginal distribution of the observations is absolutely continuous, as a weighted sum of nnth powers of eigenvalues of a certain Fredholm kernel. This enables good approximations for the joint distribution when nn repeated integrals is not a practical solution.