Product-convolution of R-symmetric unimodal distributions: an analogue of Wintner’s theorem. (English) Zbl 1420.60021
Summary: B. V. Gnedenko and A. N. Kolmogorov in their acclaimed monograph [Предельные распределения для сумм независимых случайных величин (Russian). Moskva-Leningrad: Gosudarstv. Izdat. Tekhn.-Teor. Lit. (1949; Zbl 0056.36001)] 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 A. Wintner’s [Asymptotic distributions and infinite convolutions. Ann Arbor, MI: Edwards Brothers (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 [loc. cit.] based on the relatively recent notion of R-symmetry [G. S. Mudholkar and H. Wang, J. Stat. Plann. Inference 137, No. 11, 3655–3671 (2007; Zbl 1123.60008)] is offered by showing that the product-convolutions of R-symmetric unimodal distributions are R-symmetric unimodal.
MSC:
| 60E05 | Probability distributions: general theory |
Keywords:
Wintner’s theorem; Khintchine’s characterization; inverse Gaussian distribution; root reciprocal IGReferences:
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