KMT coupling for random walk bridges

• 成果类型: Article
• 署名作者: Dimitrov, Evgeni; Wu, Xuan
• 署名单位: Columbia University; University of Chicago
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01030-y
• 发表日期: 2021
• 页码: 649-732
• 关键词: strong approximation; partial sums; fluctuations; tusnady; komlos

摘要

In this paper we prove an analogue of the Komlos-Major-Tusnady (KMT) embedding theorem for random walk bridges. The random bridges we consider are constructed through random walks with i.i.d jumps that are conditioned on the locations of their endpoints. We prove that such bridges can be strongly coupled to Brownian bridges of appropriate variance when the jumps are either continuous or integer valued under some mild technical assumptions on the jump distributions. Our arguments follow a similar dyadic scheme to KMT's original proof, but they require more refined estimates and stronger assumptions necessitated by the endpoint conditioning. In particular, our result does not follow from the KMT embedding theorem, which we illustrate via a counterexample.