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Some ARMA models for dependent sequences of Poisson counts. (English) Zbl 0664.62089

The author studies discrete-variate AR and MA models in discrete time, and having stationary Poisson-distributed marginals. The paper is a sequel to a similar paper by the author [ibid. 18, 679-705 (1986; Zbl 0603.62100)], where geometric and negative binominal marginals were considered. In equations like \((1)\quad X_ t=d^*X_{t-1}+W_ t\) the operator * stands for ‘binomial thinning’ defined by \(\alpha *N=Y_ 1+...+Y_ N\) where \(Y_ 1,Y_ 2,..\). are i.i.d. binomial with mean \(\alpha\).
Keeping in mind that in equations like (1) the thinning operations are independent for differnt t, the author computes finite dimensional distributions and covariance functions, and considers asymptotic properties.
The second part of the paper is concerned with the vector-valued AR(1)- process. Here an operation A*\(\vec X\) (matrix times random vector) is introduced in much the same way as in a paper by K. van Harn and the reviewer [Commun. Stat., Stochastic Models 2, 161-169 (1986; Zbl 0603.60012)]. Attention is focussed on the two-dimensional case. In the final part some extensions and generalizations are considered.
Reviewer: F.W.Steutel

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E15 Exact distribution theory in statistics
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