An analogue Fréchet-Shohat moments convergence theorem for indeterminate moment problems

The Fréchet-Shohat Theorem (FST) (1931) is a well-established result concerning determinate moment problems. We endeavor to address an analogous problem within the realm of indeterminate Hamburger and Stieltjes moment problems, focusing exclusively on absolutely continuous random variables. We demonstrate that, under an additional condition stipulating the convergence of the entropy of a sequence of distribution functions to the entropy of the unique maximum entropy distribution, a stronger mode of convergence is achieved, which subsequently implies convergence in distribution. This result is attainable due to the inherent property of indeterminate moment problems possessing a unique density ghmax, distinguishable from other solutions by its maximal entropy. In conclusion, the moment convergence implies weak convergence under determinacy and under indeterminacy, if there is entropy convergence to the ghmax then there is also weak convergence.