We analyze and identify stationary fields with linear regressions and quadratic conditional variances. We give sufficient conditions to determine one-dimensional distributions uniquely as normal and as certain compactly supported distributions. Our technique relies on orthogonal polynomials, which under our assumptions turn out to be a version of the so-called continuous q-Hermite polynomials.
"Stationary random fields with linear regressions." Ann. Probab. 29 (1) 504 - 519, February 2001. https://doi.org/10.1214/aop/1008956342