On Mixture, Quasi-mixture and Nearly Normal Random Processes

Abstract

The properties of both a mixture random process, specified by the multi-dimensional simple mixtures, $\hat{F}(x_1, x_2, \cdots, x_n) = a_1F_1(x_1, x_2, \cdots, x_n) + a_2F_2(x_1, x_2, \cdots, x_n), a_1 + a_2 = 1$, and a related quasi-mixture process are investigated. It is shown that if the set of random variables of the component cdf's (cumulative distribution functions) are independent, then the random variables of the resulting mixture are independent if and only if the mixture cdf $\hat{F}$ is degenerate. The quasi-mixture process, on the other hand, does have the property that factorization of the component cdf's implies factorization of the resulting mixture cdf. Specializing to the case of Gaussian cdf's, it is further shown that the GMP (Gaussian Mixture Process) never satisfies the strong mixing condition, while with reasonable assumptions on the component correlation functions the GQMP (Gaussian Quasi-Mixture Process) does satisfy the strong mixing condition. These, and other properties of the resulting mixture cdf's are of importance when mixture processes are used as models in various estimation and hypothesis testing problems. Some examples are also given for generating GMP and GQMP processes.