Characterization of Probability Distributions Via Semi-Invariants and/or Moments

While “All roads lead to Rome”, it is even more true that “Several important roads in Modern Probability Theory originate from works by P.L. Chebyshev and A.A. Markov”. Thus traditionally the moments of a distribution, and historically later, the semi- invariants (cumulants), are involved in both to characterize the distribution uniquely and use such a property to establish limit theorems. Some results, perhaps new and not so well- known, will be reported. It will be shown, based entirely on the semi-invariants, that centred and normalized sums of a specific sequence of independent and bounded random variables converge in distribution to a bounded random variable, not as one may expect, to a normal random variable. Recent general results on the moment uniqueness or non-uniqueness of both absolutely continuous and discrete distributions will be presented. A few open questions will be outlined, hoping this to be followed by discussions and welcomed suggestions.