On the Li--Zheng theorem

Abstract

By the well-known I.Kotlarski lemma, if $\xi_1$, $\xi_2$, and $\xi_3$ are independent real-valued random variables with nonvanishing characteristic functions, $L_1=\xi_1-\xi_3$ and $L_2=\xi_2-\xi_3$, then the distribution of the random vector $(L_1, L_2)$ determines the distributions of the random variables $\xi_j$ up to shift. Siran Li and Xunjie Zheng generalized this result for the linear forms $L_1=\xi_1+a_2\xi_2+a_3\xi_3$ and $L_2=b_2\xi_2+b_3\xi_3+\xi_4$ assuming that all $\xi_j$ have first and second moments, $\xi_2$ and $\xi_3$ are identically distributed, and $a_j$, $b_j$ 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 $p\ne 2$, and in the discrete field of rational numbers.