Issues on multidimensional stochastic realization

It is well known that given the (partial) correlation sequence from a 2nd order stationary Gaussian process, one can associate a (rational) positive real function with it. Indeed, a partial correlation sequence can be essentially be described by a lossless two-port, a passive loading of which provides all extensions of partial correlation sequence. While the above is true for one-dimensional processes (1D), a similar treatment becomes problematic in two or higher dimensions. The source of this difficulty is closely related to the difficulties associated with extending classic passive circuit synthesis methods to two or higher dimensions. The trigonometric moment problem also suffers from similar difficulties when a multidimensional extension is sought. The paper will explore in some detail the parallel between these closely related problems.