  We study a collection of discrete Markov chains related to the causal set
approach to modeling discrete theories of quantum gravity. The transition
probabilities of these chains satisfy a general covariance principle, a
causality principle, and a renormalizability condition. The corresponding
dynamics are completely determined by a sequence of nonnegative real coupling
constants. Using techniques related to the classical moment problem, we give a
complete description of any such sequence of coupling constants. We prove a
representation theorem: every discrete theory of quantum gravity arising from
causal set dynamics satisfying covariance, causality and renormalizability
corresponds to a unique probability distribution function on the nonnegative
real numbers, with the coupling constants defining the theory given by the
moments of the distribution.
