Symmetric semistable measures on Rn and symmetric semistable distribution operators in quantum mechanics

The paper contains two theorems. The first one deals with the classic probability, provides the form of the characteristic function of a symmetric, semistable probability distribution in Rⁿ. By a semistable probability distribution we mean a certain class of limit distributions of the sequence of random variables, containing the classic stable probability distribution and included in a set of infinitely divisible distributions.The result will be applied in the proof of the second theorem on non-commutative probability, in which we deal with the generalization of the central limit theorem in quantum mechanics.In the paper we define the symmetric semistable probability distributions for pairs of canonical observables in a state ?₀. In the second theorem we indicate that the symmetric semistable distribution operator is a ground state. By a ground state we mean a state for which Heisenberg's inequality becomes an equality.