On the Mixture of Distributions

If F = \ is a family of distribution functions and μ is a measure on a Borel Field of subsets of F with μ(F) = 1, then ∫ F(·) dμ (F) is again a distribution function which is called a μ-mixture of F. In Section 2, convergence questions when either Fₙ or μₖ (or both) tend to limits are dealt with in the case where F is indexed by a finite number of parameters. In Part 3, mixtures of additively closed families are considered and the class of such μ-mixtures is shown to be closed under convolution (Theorem 3). In Section 4, a sufficient as well as necessary conditions are given for a μ-mixture of normal distributions to be normal. In the case of a product-measure mixture, a necessary and sufficient condition is obtained (Theorem 7). Generation of mixtures is discussed in Part 5 and the concluding remarks of Section 6 link the problem of mixtures of Poisson distributions to a moment problem.