Effective Laplace inversion

• 成果类型: Article
• 署名作者: Stef, A; Tenenbaum, G
• 署名单位: Universite de Lorraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2001
• 页码: 558-575
• 关键词: gaussian law; errors

摘要

Let F, G be arbitrary distribution functions on the real line and let F, G denote their respective bilateral Laplace transforms. Let K > 0 and let h: R+ --> R+ be continuous, non-decreasing, and such that h(u) greater than or equal to Au-4 for some A > 0 and all u > 0. Under the assumptions that sup \(F) over cap (u)- (G) over cap (u)\ less than or equal to epsilon , (F) over cap (u)+ (G) over cap (u) less than or equal to h(u) (-L less than or equal to u less than or equal to L), 0 less than or equal to u less than or equal to kappa we establish the bound sup \(F) over cap (u) - (G) over cap (u)\ less than or equal to CQ(G)(l) u epsilon R where C is a constant depending at most on kappa and A, Q(G) is the concentration function of G, and l := (log L)/L + (log W)/W, with W any solution to h(W) = 1/epsilon. Improving and generalizing an estimate of Alladi, this result provides a Laplace transform analogue to the Berry-Esseen inequality, related to Fourier transforms. The dependence in epsilon is optimal up to the logarithmic factor log W. A number-theoretic application, developed in detail elsewhere, is described. It concerns so-called lexicographic integers, whose characterizing property is that their divisors are ranked according to size and valuation of the largest prime factor. The above inequality furnishes, among other informations, an effective Erdos-Kac theorem for lexicographical integers.