This paper describes a new method for generating stationary integervalued time series from renewal processes. We prove that if the lifetime distribution of renewal processes is nonlattice and the probability generating function is rational, then the generated time series satisfy causal and invertible ARMA type stochastic difference equations. The result provides an easy method for generating in...

We have derived some matrix equations for speedy computation of the conditional covariance kernel of a discrete-time process obtained from irregularly sampling an underlying continuous-time ARMA process. These results are applicable to both stationary and non-stationary ARMA processes. We have also demonstrated that these matrix results can be useful in shedding new insights on the covariance s...

This paper discusses the stochastic process structure of certain differential transformations (OTis) associated with perfectly observed ARMA processes and uses DT's to obtain the asymptotic information matrix for possibly non-Gaussian situations. The DT's can also be applied to implement approximate M-estimate algorithms for the ARMA model parameters. M-estimates yield asymptotic efficieQcy rob...

Nonstationary ARIMA processes and nearly nonstationary ARMA processes, such as autoregressive processes having a root of the AR polynomial close to the unit circle, have sample autocovariance and spectral properties that are, in practice, almost indistinguishable from those of a stationary longmemory process, such as a Fractionally Integrated ARMA (ARFIMA) process. Because of this, model misspe...

Second order properties of nearly nonstationary ARMA processes are investigated in the cases when the autoregressive polynomial equation has (i) a real root close to 1; (ii) a real root close to -1; (iii) a pair of complex roots close to the unit circle. The effect of the closeness to the unit circle of the ARMA poles on its covariance and spectral density functions is considered. The obtained ...

Gaussian ARMA processes with continuous time parameter, otherwise known as stationary continuous-time Gaussian processes with rational spectral density , have been of interest for many years. In the last twenty years there has been a resurgence of interest in continuous-time processes, partly as a result of the very successful application of stochastic diierential equation models to problems in...