  Large-time correlation functions have a pivotal role in extracting particle
masses from Euclidean lattice field theory calculations, however little is
known about the statistical properties of these quantities. In this work, the
asymptotic form of the distributions of the correlation functions at vanishing
momentum is determined for bosonic interacting lattice field theories with a
unique gapped vacuum. It is demonstrated that the deviations from the
asymptotic form at large Euclidean times can be utilized to determine the
spectrum of the theory.
