## Non Gaussian autoregressive and moving average schemes.(English)Zbl 0946.62086

Szyszkowicz, Barbara (ed.), Asymptotic methods in probability and statistics. A volume in honour of Miklós Csörgő. ICAMPS ’97, an international conference at Carleton Univ., Ottawa, Ontario, Canada, July 1997. Amsterdam: North-Holland/ Elsevier. 731-737 (1998).
Introduction: Consider the system of equations $\sum^p_{k= 0}\widetilde\phi_k x_{t-k}= \sum^q_{j= 0}\widetilde\theta_j \xi_{t-j},$ where $$\xi_t$$’s are independent, identically distributed random variables with $$E\xi_t\equiv 0$$, $$E\xi^2_t\equiv 1$$. Let $$\widetilde\phi_0= \widetilde\theta_0= 1$$ and $$\widetilde\phi_p, \widetilde\theta_q\neq 0$$. The polynomials $\phi(z)= \sum^p_{k= 0} \widetilde\phi_k z^k,\quad \theta(z)= \sum^q_{j= 0} \widetilde\theta_j z^j,$ are assumed to have no roots in common. There is then a stationary solution $$\{x_t\}$$ of the system if and only if $$\phi(z)$$ has no roots of absolute value one and this solution is uniquely determined. The solution is called an autoregressive moving average (ARMA) process. If $$\phi(z)\equiv 1$$ the solution is a Moving Average (MA) while if $$\theta(z)\equiv 1$$ it is an Autoregressive Scheme (AR). If all the roots of $$\phi(z)$$ and $$\theta(z)$$ are outside the unit disc $$|z|\leq 1$$ in the complex plane we shall call the solution minimum phase.
It is well known that in the case of a stationary ARMA minimum phase process, the best predictor of $$x_t$$ in mean square given the past is linear. A necessary condition for the best predictor to be linear is given in terms of a functional equation. This condition is then used in the case of autoregressive schemes and moving averages. A simple example is given of a nonminimum phase scheme in which the best predictor is nonlinear.
For the entire collection see [Zbl 0901.00049].

### MSC:

 62M20 Inference from stochastic processes and prediction