Conditional orthogonality and conditional stochastic realization

The concept of conditional orthogonality for the random variables x, y with respect to a third random variable z is extended to the case of a triple x, y, z of processes and is shown to be equivalent to the property that the space spanned by the conditioning process z splits the spaces generated by the conditionally orthogonal processes x, y. The main result is that for jointly wide sense stationary processes x, y, z, conditional orthogonality plus a strong feedback free condition on (z, x) and (z, y), or, equivalently, splitting plus this condition, is equivalent to the existence of a stochastic realization for the joint process (x, y, z) in the special class of so-called conditionally orthogonal stochastic realizations.