DISCREPANCY BOUNDS FOR A CLASS OF NEGATIVELY DEPENDENT RANDOM POINTS INCLUDING LATIN HYPERCUBE SAMPLES

• 成果类型: Article
• 署名作者: Gnewuch, Michael; Hebbinghaus, Nils
• 署名单位: University Osnabruck; University of Kiel
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1638
• 发表日期: 2021
• 页码: 1944-1965
• 关键词: monte-carlo variance; error-bounds; star-discrepancy; integration; CONSTRUCTION; tractability; variables; numbers

摘要

We introduce a class of gamma-negatively dependent random samples. We prove that this class includes, apart from Monte Carlo samples, in particular Latin hypercube samples and Latin hypercube samples padded by Monte Carlo. For a gamma-negatively dependent N-point sample in dimension d we provide probabilistic upper bounds for its star discrepancy with explicitly stated dependence on N, d, and gamma. These bounds generalize the probabilistic bounds for Monte Carlo samples from Heinrich et al. (Acta Arith. 96 (2001) 279-302) and C. Aistleitner (J. Complexity 27 (2011) 531-540), and they are optimal for Monte Carlo and Latin hypercube samples. In the special case of Monte Carlo samples the constants that appear in our bounds improve substantially on the constants presented in the latter paper and in C. Aistleitner and M. T. Hofer (Math. Comp. 83 (2014) 1373-1381).