On the Li--Zheng theorem

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 $p\mathrm{\ne }2$, and in the discrete field of rational numbers.