In the present note we show that Polya’s type characterization theorem of Gaussian distributions does not hold. This happens because in the linear form, constituted by the independent copies of quaternion random variables, a part of the quaternion coefficients is written on the right hand side and another part on the left side. This gives a negative answer to the question posed in [1]. Mathemat...

The notion of a proposition as a set of possible worlds or states occupies central stage in probability theory, semantics and epistemology, where it serves as the fundamental unit both of information and meaning. But this fact should not blind us to the existence of prospects with a di¤erent structure. In the paper I examine the use of random variables in particular, proposition-valued random v...

We obtain asymptotic expansions for local probabilities of partial sums uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The involve products polynomials and trigonometric polynomials. These results also have counterparts triangular arrays. Our do require any additional assumptions. As an application our we find necessary sufficient cond...