The existence of an asymptotic shape at large times of the probability density function (PDF) of a non-reacting scalar, mixed by a solenoidal turbulent velocity field and molecular diffusive transport, was investigated by Sinai and Yakhot [Y. G. Sinai and V. Yakhot, “Limiting probability distributions of a passive scalar in a random velocity field,” Phys. Rev. Lett. 63, 1962 (1989)]. The quasi-stationarity of the mixing statistics along the time evolution by Valiño et al. [“Quasistationary probability density functions in the turbulent mixing of a scalar field,” Phys. Rev. Lett. 72, 3518 (1994)] was an extension to symmetric scalar pdfs; analytic solutions for the scalar fluctuation dissipation rates, conditional upon the scalar value, and the pdf were obtained. This manuscript examines the generalization of the latter results to asymmetric scalar pdfs, further scrutinizes underlying mechanisms for a quasi-stationary statistics, and shows a Monte Carlo implementation which allows non-Gaussian relaxations to prescribed values of skweness and kurtosis.