a1 Princeton University
a2 UC San Diego
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.
We thank Stéphane Bonhomme, Susanne Schennach, the editor, the co-editor, and two anonymous referees for helpful comments. Evdokimov gratefully acknowledges the support from the Gregory C. Chow Econometric Research Program at Princeton University.