Failure of almost uniformly convergence for noncommutative martingales

2024/02/13 by Guixiang Hong, Hong, Guixiang, Éric Ricard +1
Decision Sciences · Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.2402.08560

openalex publication_date 2024/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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