On transformations of multivariate ARMA processes

Let Xt be an /-dimensional ARMA (p, q) process. Let g: U l -> W be a measurable function. Define a process Zt by Zt = g(Xt). Generally, Z.is not an ARMA process. However, we are interested in such functions g, for which Zt is also an AR process. It is important to know the orders of the process Zt. In the most cases we find only some bounds for them. From the practical point of view, our considerations enable to express complicated ARMA processes in a form of some transformations of simpler ARMA processes. The problem of transformations of one-dimensional ARMA processes was investigated in several papers. The results concern mainly sum, product and aggregation of ARMA processes. A unified approach was presented by Engel [2]. It is based on a theorem which characterizes a one-dimensional ARMA process using a property of its covariance function. In Engel's paper also all other references can be found. There exist only few papers devoted to transformations of multidimensional ARMA processes. They concern mainly the sum of ARMA processes. Let us mention the paper by Lutkepohl [8], who considers a transformation Yt = FX, where F is a matrix with real elements. In our paper we prove some assertions on the scalar product of two ARMA processes. The methods are based on Theorem 3.1, which characterizes a multivariate ARMA process by means of its covariance function. Theorem 3.1 is a generalization of the assertion for one-dimensional processes given in [2]. The problems concerning bounds for orders of the model are demonstrated on some examples.