On a decomposition problem for multivariate probability measures

Abstract

The aim of this paper is to describe the equivalence classes (e.c.) of the following equivalence relation on the set P n of probability measures on R n : μ ∼ ν if μ ∗ μ̄ = ν ∗ ν̄, where μ̄( A) = μ(− A) for Borel sets A ⊂ R n . μ ∼ ν iff | φ μ ( t)| = | φ ν ( t)| for t ∈ R n , where φ μ is the characteristic function of μ. Sufficient conditions for the triviality of an e.c. (i.e., up to translations the e.c. consists of a unique pair (μ, μ̄)) are given. Gaussian measures are the only infinitely divisible probability measures whose e.c.'s are trivial. Distinct symmetrical probability measures with equal moments of all orders may have characteristic functions with the same absolute value. The corresponding problem is also considered on R n / Z n .