  Motivated by the dynamic modeling of relative abundance data in ecology, we
introduce a general approach to model stationary Markovian or non Markovian
time series on (relatively) compact spaces such as a hypercube, the simplex or
a sphere in the Euclidean space. Our approach is based on a general
construction of infinite memory models, called chains with complete
connections. The two main ingredients involved in our generic construction are
a parametric family of probability distributions on the state space and a map
from the state space to the parameter space. Our framework encompasses
Markovian models, observation-driven models and more general infinite memory
models. Simple conditions ensuring the existence and uniqueness of a stationary
and ergodic path are given. We then study in more details statistical inference
in two time series models on the simplex, based on either a Dirichlet or a
multivariate logistic-normal conditional distribution. Usefulness of our models
to analyze abundance data in ecosystems is also discussed.
