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ISSN 1547-7363(online) ISSN 0094-9000(print)
A Markovian Gauss inequality for asymmetric deviations from the mode of symmetric unimodal distributions
Author:
Chris A.J. Klaassen
Journal:
Theor. Probability and Math. Statist. 111 (2024), 9-19
MSC (2020):
Primary 60E15
DOI:
https://doi.org/10.1090/tpms/1215
Published electronically:
October 30, 2024
Full-text PDF
Abstract | References | Similar Articles | Additional Information
For a random variable with a unimodal distribution and finite second moment Gauß (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it under the assumption of a finite $r$-th absolute moment is given ($r\geq 1$), based on the Khintchine representation of unimodal random variables. A special instance of the resulting Narumi–Gauß inequality is the one with finite first absolute moment, which might be called a Markovian Gauß inequality.
For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized the Gauß inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.
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Additional Information
Chris A.J. Klaassen
Affiliation:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands
Address at time of publication:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands
Email:
c.a.j.klaassen@uva.nl
Keywords:
Markovian Gauß inequality,
Khintchine representation,
Volkov inequality
Received by editor(s):
November 3, 2023
Accepted for publication:
February 28, 2024
Published electronically:
October 30, 2024
Article copyright:
© Copyright 2024
Taras Shevchenko National University of Kyiv