I recently read You Cannot Recover a Distribution From Its Moments Alone. Stieltjes Proved This in 1894. Every Moment-Matching Method in Your ML Pipeline Has a Silent Failure Mode.

Back in 1972, in my first serious course on probablility, I leaarned about moment generating functions, with a warning that they are not as powerful as characteristic functions. In fact they may not exist. The characteristic function always does. The characteristic function is simply the Fourier Transform of the probability density function. Fourier Transforms I encountered many times in my physics studies. Given the requirements of a probability density function, the FT always exists and can be applied to ML problems.

As the articles describes, the characteristic function leads to a clear proof of the Central Limit theorem. It explicitly shows how the key assumption of the CLT, finite variance, appears in the derivation.