This note demonstrates that the conditions of Kotlarski’s (1967, <italic>Pacific Journal of Mathematics</italic> 20(1), 69–76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of <italic>M</italic>, <italic>U</italic> <sub>1</sub>, and <italic>U</italic> <sub>2</sub> are nonvanishing can be replaced with much weaker conditions: The characteristic function of <italic>U</italic> <sub>1</sub> can be allowed to have real zeros, as long as the derivative of its characteristic function at those points is not also zero; that of <italic>U</italic> <sub>2</sub> can have an isolated number of zeros; and that of <italic>M</italic> need satisfy no restrictions on its zeros. We also show that Kotlarski’s lemma holds when the tails of <italic>U</italic> <sub>1</sub> are no thicker than exponential, regardless of the zeros of the characteristic functions of <italic>U</italic> <sub>1</sub>, <italic>U</italic> <sub>2</sub>, or <italic>M</italic>.
