  This paper studies theory and inference related to a class of time series
models that incorporates nonlinear dynamics. It is assumed that the
observations follow a one-parameter exponential family of distributions given
an accompanying process that evolves as a function of lagged observations. We
employ an iterated random function approach and a special coupling technique to
show that, under suitable conditions on the parameter space, the conditional
mean process is a geometric moment contracting Markov chain and that the
observation process is absolutely regular with geometrically decaying
coefficients. Moreover the asymptotic theory of the maximum likelihood
estimates of the parameters is established under some mild assumptions. These
models are applied to two examples; the first is the number of transactions per
minute of Ericsson stock and the second is related to return times of extreme
events of Goldman Sachs Group stock.
