Sharpness of Lenglart’s domination inequality and a sharp monotone version
- Authors: Sarah Geiss, Michael Scheutzow
- Preprint year: 2021
- First public date: 2021-01-26
- arXiv: 2101.10884
- Status: Published
- Publication type: Journal article
- Publication year: 2021
- Journal: Electronic Communications in Probability 26, 1–8 (2021)
- DOI: 10.1214/21-ECP413
Abstract
We prove that the best so far known constant $c_p=\frac{p^{-p}}{1-p},\, p\in(0,1)$ of a domination inequality, which originates to Lenglart, is sharp. In particular, we solve an open question posed by Revuz and Yor. Motivated by the application to maximal inequalities, like e.g. the Burkholder-Davis-Gundy inequality, we also study the domination inequality under an additional monotonicity assumption. In this special case, a constant which stays bounded for $p$ near $1$ was proven by Pratelli and Lenglart. We provide the sharp constant for this case.