Econometric Theory

Table of Contents - August 2012 - Volume 28, Issue 04  

NOTES AND PROBLEMS

SOME EXTENSIONS OF A LEMMA OF KOTLARSKI

Kirill Evdokimova1 c1 and Halbert Whitea2

a1 Princeton University

a2 UC San Diego

Abstract

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.

Correspondence:

c1 Address correspondence to Kirill Evdokimov, Department of Economics, Princeton University, Fisher Hall 001, Princeton, NJ 08544; e-mail: kevdokim@princeton.edu.

Footnotes

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.