  Mixture transition distribution time series models build high-order
dependence through a weighted combination of first-order transition densities
for each one of a specified number of lags. We present a framework to construct
stationary transition mixture distribution models that extend beyond linear,
Gaussian dynamics. We study conditions for first-order strict stationarity
which allow for different constructions with either continuous or discrete
families for the first-order transition densities given a pre-specified family
for the marginal density, and with general forms for the resulting conditional
expectations. Inference and prediction are developed under the Bayesian
framework with particular emphasis on flexible, structured priors for the
mixture weights. Model properties are investigated both analytically and
through synthetic data examples. Finally, Poisson and Lomax examples are
illustrated through real data applications.
