We consider bivariate densities having diagonal expansions and review and generalize some of its known properties. In particular, Mehler's equality and Gebelein's inequality are generalized. Moreover, we consider stationary processes   with a covariance function r(i) and with bivariate densities of (X₁, X_1+i) having diagonal form with coefficients a_k(i), k = 0,1,… and state general conditions under which sequences subordinated to (X_i)^∞_i=1 are long-range dependent and obey the reduction principle. Furthermore, in the special case a_k(i) = r(i)^k, k = 0,1,… estimates based on such sequences enjoy some common asymptotic properties under long-range dependence.