Stationary Markov chains with linear regressions

In Bryc (Ann. Probab., (1998), to appear), we determined one-dimensional di stributions of a stationary field with linear regressions (1) and quadratic conditional variances (2) under a linear constraint (7) on the coefficient s of the quadratic expression (3). In this paper, we show that for stationa ry Markov chains with linear regressions and quadratic conditional variance s the coefficients of the quadratic expression are indeed tied by a linear constraint which can take only one of the two alternative forms (7), or (8) . (C) 2001 Elsevier Science B.V. All rights reserved.