Failure of almost uniformly convergence for noncommutative martingales

Guixiang Hong; Éric Ricard

Failure of almost uniformly convergence for noncommutative martingales

Operator Algebras 2024-07-09 v3 Functional Analysis

Authors: Guixiang Hong , Éric Ricard

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 $1\leq p<2$ . 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.

Keywords

noncommutative martingales almost uniform convergence ergodic averages distributional function transference techniques

Cite

@article{arxiv.2402.08560,
  title  = {Failure of almost uniformly convergence for noncommutative martingales},
  author = {Guixiang Hong and Éric Ricard},
  journal= {arXiv preprint arXiv:2402.08560},
  year   = {2024}
}

Comments

7 pages, final version incorporating referees' comments, to appear in Probability Theory and Related Fields

Failure of almost uniformly convergence for noncommutative martingales

Abstract

Keywords

Cite

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