Abstract:We show that the characteristic function of a unimodal probability distribution function can be inverted by the Fourier transform a.e. if and only if the distribution is absolutely continuous. The result complements Khintchine’s criterion for unimodal distributions.
- E. M-J. Bertin, I. Cuculescu, and R. Theodorescu, Unimodality of Probability Measures, Kluwer Academic Publishers, 1997.
- Sudhakar Dharmadhikari and Kumar Joag-Dev, Unimodality, convexity, and applications, Probability and Mathematical Statistics, Academic Press, Inc., Boston, MA, 1988. MR 954608
- Hugo Aimar and Liliana Forzani, Weighted weak type inequalities for certain maximal functions, Studia Math. 101 (1991), no. 1, 105–111. MR 1141366, DOI 10.4064/sm-101-1-105-111
- Richard R. Goldberg, Fourier transforms, Cambridge Tracts in Mathematics and Mathematical Physics, No. 52, Cambridge University Press, New York, 1961. MR 0120501
- Tatsuo Kawata, Fourier analysis in probability theory, Probability and Mathematical Statistics, No. 15, Academic Press, New York-London, 1972. MR 0464353
- A. Ya. Khintchine, On Unimodal Distributions, Izv. Nauchno Issled. Inst. Mat. Mech. Temsk. Gos. Univ. 2, no. 2, 1–7 (1938).
- Eugene Lukacs, Characteristic functions, Hafner Publishing Co., New York, 1970. Second edition, revised and enlarged. MR 0346874
- Pál Medgyessy, Decomposition of superposition of density functions on discrete distribution, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 21 (1973), 129–200 (Hungarian). MR 0440659
- Constantine Georgakis
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
- Received by editor(s): December 14, 1994
- Received by editor(s) in revised form: March 10, 1997
- Communicated by: J. Marshall Ash
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3239-3241
- MSC (1991): Primary 42A38; Secondary 60E10
- DOI: https://doi.org/10.1090/S0002-9939-98-04385-8
- MathSciNet review: 1452804