  We give the cumulative distribution function of $M_n$, the maximum of a
sequence of $n$ observations from an ARMA(1, 1) process. Solutions are first
given in terms of repeated integrals and then for the case, where the
underlying random variables are absolutely continuous. The distribution of
$M_n$ is then given as a weighted sum of the $n$th 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.
  These results are large deviations expansions for estimates, since the
maximum need not be standardized to have a limit. In fact, such a limit need
not exist.
