  Auto-correlation functions in an equilibrium stochastic process are
well-characterized by Bochner's theorem as Fourier transforms of a finite
symmetric Borel measure. The existence of a long-time limit of these
correlation functions depends on the spectral properties of the measure. Here
we provide conditions applicable to a wide-class of dynamical theories
guaranteeing the existence of the long-time limit. We discuss the implications
in the context of the mode-coupling theory of the glass transition where a
non-trivial long-time limit signals an idealized glass state.
