Weighted inequalities for the martingale square and maximal functions

Adam Osȩkowski

Statistics & Probability Letters, 2017, vol. 120, issue C, 95-100

Abstract: Let W be a weight, i.e., a uniformly integrable, continuous-path martingale, and let W∗ denote the associated maximal function. We show that if X is an arbitrary càdlàg martingale and X∗, [X] denote its maximal and square functions, then ‖[X]1/2‖Lp(W)≤γp‖X∗‖Lp(W∗),1≤p≤2, where γp2=1+supt>1(2t−1)(1−tp−2)tp−1. The estimate is sharp for p∈{1,2}. Furthermore, it is proved that if p>2, then the above weighted inequality does not hold with any finite constant γp depending only on p.

Keywords: Martingale; Maximal function; Square function; Best constant (search for similar items in EconPapers)
Date: 2017
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0167715215300146
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:120:y:2017:i:c:p:95-100

Ordering information: This journal article can be ordered from
http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spl.2016.09.020

Access Statistics for this article

Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul

More articles in Statistics & Probability Letters from Elsevier
Bibliographic data for series maintained by Catherine Liu ().