Notes and problems some extensions of a lemma of kotlarski

This note demonstrates that the conditions of Kotlarski's (1967, Pacific Journal of Mathematics 20(1), 69-76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of M, U ₁, and U ₂ are nonvanishing can be replaced with much weaker conditions: The characteristic function of U ₁ 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 U ₂ can have an isolated number of zeros; and that of M need satisfy no restrictions on its zeros. We also show that Kotlarski's lemma holds when the tails of U ₁ are no thicker than exponential, regardless of the zeros of the characteristic functions of U ₁, U ₂, or M.