Some results for $U$-statistics and strong demimartingales

Decheng Feng; Xiaomei Zhang

doi:10.4134/BKMS.b230377

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2024; 61(5): 1211-1221

Online first article September 23, 2024 Printed September 30, 2024

https://doi.org/10.4134/BKMS.b230377

Some results for $U$-statistics and strong demimartingales

Decheng Feng, Xiaomei Zhang

Northwest Normal University; Northwest Normal University

Abstract

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Abstract

In this paper, we obtain a result that a $U$-statistic $U_n$ based on associated random variables and a kernel $k(x_1,\ldots,x_m)$ which is a bounded variance function can be written as the difference of two strong demimartingales. Moreover, we establish Chow type minimal inequalities for strong demimartingales and obtain Marshall type minimal inequalities as well as a Value-at-Risk bound for ($\mathop{\min}_{1\leq k \leq n}S_k$).

Keywords: Minimal inequality, $U$-statistic, strong demimartingales, associated random variables

MSC numbers: Primary 60E15, 60G48

Supported by: This work was financially supported by National Natural Science Foundation of China (Grant Nos: 12261080; 62261049).