On the Mixture of Distributions

Abstract

If $\mathcal{F} = \{F\}$ is a family of distribution functions and $\mu$ is a measure on a Borel Field of subsets of $\mathcal{F}$ with $\mu(\mathcal{F}) = 1$, then $\int F(\cdot) d\mu (F)$ is again a distribution function which is called a $\mu$-mixture of $\mathcal{F}$. In Section 2, convergence questions when either $F_n$ or $\mu_k$ (or both) tend to limits are dealt with in the case where $\mathcal{F}$ is indexed by a finite number of parameters. In Part 3, mixtures of additively closed families are considered and the class of such $\mu$-mixtures is shown to be closed under convolution (Theorem 3). In Section 4, a sufficient as well as necessary conditions are given for a $\mu$-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.