  We examine convergence properties of reconstructing the generalized parton
distributions (GPDs) through the universal moment parameterization (GUMP). We
provide a heuristic explanation for the connection between the formal
summation/expansion and the Mellin-Barnes integral in the literature, and
specify the exact convergence condition. We derive an asymptotic condition on
the conformal moments of GPDs to satisfy the boundary condition at $x=1$ and
subsequently develop an approximate formula for GPDs when $x>\xi$. Since
experimental observables constraining GPDs can be expressed in terms of double
or even triple summations involving their moments, scale evolution factors, and
Wilson coefficients, etc., we propose a method to handle the ordering of the
multiple summations and convert them into multiple Mellin-Barnes integrals via
analytical continuations of integer summation indices.
