An Analogue Fréchet-Shohat Moments Convergence Theorem for Indeterminate Moment Problems

Originally established in 1931, the Fr\'echet-Shohat Theorem (FST) is a fundamental result in the method of moments. It provides sufficient conditions under which the convergence of a sequence of moments $\{m_{k, n}\}_{n=1}^{\infty}$ to a limit sequence $\{m_k\}$ ensures the weak convergence of the associated distribution functions $\{G_n\}$ to a limit $G$. A critical requirement of the classical theorem is the determinacy of the underlying moment problem; that is, $G$ must be the unique distribution characterized by $\{m_k\}$. This study extends the foundational Fr\'echet-Shohat framework to the setting of indeterminate Hamburger and Stieltjes moment problems, where the classical FST traditionally fails due to the non-uniqueness of the limiting distribution. We demonstrate that by imposing an entropic constraint on the sequence $\{G_n\}$ - specifically, convergence in Shannon entropy - one can recover a unique limit entropy-distinguishable distribution, $G_{hmax}$, from the indeterminate class having the given moments. This result facilitates a unified treatment of FST across both determinate and indeterminate frameworks.