Classes of measures closed under mixing and convolution. Weak stability
For a random vector X with a fixed distribution μ we construct a class of distributions M(μ) = {μ ◦ λ : λ ∈ P}, which is the class of all distributions of random vectors XΘ, where Θ is independent of X and has distribution λ. The problem is to characterize the distributions μ for whichM(μ) is closed under convolution. This is equivalent to the characterization of the random vectors X such that for all random variables Θ1, Θ2 independent of X,X ′ there exists a random variable Θ independent of X such that