Fréchet–Shohat moments convergence theorem for discrete distributions with non-negative support: Stronger modes of convergence

By using recent advances in information theory and the principle of maximum entropy (MaxEnt), alongside new results established herein regarding the entropic convergence of moment-constrained MaxEnt distributions, we demonstrate that convergence can be strengthened beyond the standard weak topology for discrete distributions with non-negative countable support. Specifically, we prove that sequence-based MaxEnt approximations converge in relative entropy to an unknown target distribution 𝐺 uniquely characterized by a sequence of moment constraints. Building upon this foundation, we analyze a sequence of cumulative distribution functions {𝐺_𝑀 }_{M=1}^{∞} supported on R^{+}, and establish sufficient conditions under which their convergence to 𝐺 occurs in a topology strictly stronger than the weak topology.