Multivariate reciprocal stationary Gaussian processes (Q1097574)

A scalar valued random process X(t), \(t\in (a,b)\), is called reciprocal if for all s,t \((a<s<t<b)\) the \(\sigma\)-fields \(\sigma\{\) X(r):r\(\in (s,t)\}\) and \(\sigma\{\) X(r):r\(\in (a,b)\setminus (s,t)\}\) are conditionally independent given X(s) and X(t). All the covariances R(t), \(t\in (a,b)\), of Gaussian stationary processes having the reciprocal property were characterized by the authors in terms of the factorization property \[ R(s-t)-R(s)R(t)=h(s)g(t) \] for suitable functions h and g in C. R. Acad. Sci., Paris, Sér. I 295, 291-293 (1982; Zbl 0506.60031). An analogon of this characterization is now presented for multivariate stationary Gaussian processes.