On the Li--Zheng theorem

By the well-known I.Kotlarski lemma, if ξ₁, ξ₂, and ξ₃ are independent real-valued random variables with nonvanishing characteristic functions, L₁=ξ₁-ξ₃ and L₂=ξ₂-ξ₃, then the distribution of the random vector (L₁, L₂) determines the distributions of the random variables ξⱼ up to shift. Siran Li and Xunjie Zheng generalized this result for the linear forms L₁=ξ₁+a₂ξ₂+a₃ξ₃ and L₂=b₂ξ₂+b₃ξ₃+ξ₄ assuming that all ξⱼ have first and second moments, ξ₂ and ξ₃ are identically distributed, and aⱼ, bⱼ satisfy some conditions. In the article, we give a simpler proof of this theorem. In doing so, we also prove that the condition of existence of moments can be omitted. Moreover, we prove an analogue of the Li--Zheng theorem for independent random variables with values in the field of p-adic numbers, in the field of integers modulo p, where p≠ 2, and in the discrete field of rational numbers.