A three-parameter family of probability distributions is constructed
such that its Mellin transform is defined over the same domain as
the 2D GMC on the Riemann sphere with three insertion points
and
satisfies the DOZZ formula in the sense of Kupiainen et al. (Ann. Math. 191 (2020)
81–166). The probability distributions in the family are defined as products of
independent Fyodorov–Bouchaud and powers of Barnes beta distributions of types
and
. In the special
case of
the constructed probability distribution is shown to be consistent with the known
small deviation asymptotic of the 2D GMC laws with everywhere-positive
curvature.