On some characterizations of the mixture of gamma distributions

Following Gupta and Wesolowski [1997. Uniform mixtures via posterior means. Ann. Inst. Statist. Math. 49, 171-180], in this work, under the condition X/U and U are independent, X/U has a Be(p,q) distribution, and given X the conditional expectation of a certain function of (U,X) is constant, we characterize the distribution of (U,X). This problem is related to Lukacs type characterization, where both X and Y have to be gamma distributed with the same scale parameter, if both X and Y, and X/(X+Y) and X+Y are independent. Among others, we prove if q=1, and for some integer n⩾1, E(∑i=1nai(U-X)i|X)=b, where a1,…,an,b, are real constants such that a12+⋯+an2≠0 and b≠0, or for some real number n>0, E((U-X)n|X)=b, where b>0 is a constant, then the distribution of (U,X) can be determined.