L1 bounds in normal approximation

• 成果类型: Article
• 署名作者: Goldstein, Larry
• 署名单位: University of Southern California
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000001123
• 发表日期: 2007
• 页码: 1888-1930
• 关键词: steins method; limit; remainder; zero

摘要

The zero bias distribution W* of W, defined though the characterizing equation EW f (W) = sigma(2) Ef ''(W*) for all smooth functions f, exists for all W with mean zero and finite variance sigma(2). For W and W* defined on the same probability space, the L-1 distance between F, the distribution function of W with EW = 0 and Var(W) = 1, and the cumulative standard normal phi has the simple upper bound parallel to F - Phi parallel to(1) <= vertical bar W-* - W vertical bar This inequality is used to provide explicit L-1 bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere S(l(n)(p)), simple random sampling and combinatorial central limit theorems.