Failure of almost uniform convergence for noncommutative martingales

• 成果类型: Article; Early Access
• 署名作者: Hong, Guixiang; Ricard, Eric
• 署名单位: Harbin Institute of Technology; Universite de Caen Normandie; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01300-5
• 发表日期: 2024
• 关键词: ergodic-theorems

摘要

In this paper, we provide a counterexample to show that in sharp contrast to the classical case, the almost uniform convergence may not happen for truly noncommutative L-p-martingales when 1 <= p < 2. The same happens to ergodic averages. The proof consists of some sharp estimates of the distributional function of a sequence of matrices and some non standard transference techniques, which might admit further applications.