Product-Convolution of R-Symmetric Unimodal Distributions: An Analogue of Wintner's Theorem

Abstract

Gnedenko and Kolmogorov (1949) in their acclaimed monograph. Limit Distributions for Sums of Independent Random Variables, claimed that the convolutions of unimodal distributions are unimodal. Kai Lai Chung, in an appendix of his English translation of the monograph, by a counterexample, refuted the claim and further noted Wintner's (1938) result that the convolutions of symmetric unimodal distributions are symmetric unimodal. In this note, it is shown that the product-convolutions of unimodal distributions are not unimodal either. Furthermore, an analogue of Wintner's result based on the relatively recent notion of R-symmetry (Mudholkar and Wang, 2007) is offered by showing that the product-convolutions of R-symmetric unimodal distributions are R-symmetric unimodal.

Key Words:

• Wintner's theorem
• Khintchine's characterization
• inverse Gaussian distribution
• Root reciprocal IG