Аннотация:
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 semiinvariants, 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 nonuniqueness 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.
