Completeness and self-decomposability of mixtures (Q1059946)

A family \(\{P_{\theta}\), \(\theta\in \Theta \}\) of probability distributions is defined to be strongly complete if \(E_{\theta}[g(X)]=0\) for all \(\theta\) in a dense subset of \(\Theta\) implies that \(g(X)=0\) a.s. \((P_{\theta})\) for all \(\theta\in \Theta\). Obviously exponential families (of standard type) are strongly complete. From the result that mixtures of strongly complete families are complete if the mixing family is complete, it is deduced that the generalized Waring distribution is complete with respect to each of its parameters. A negative result about self-decomposability of mixtures is also given.