A Uniform Random Pointwise Ergodic Theorem

2017/08/16 by Ben Krause, Krause, Ben, Pavel Zorin‐Kranich +1 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1708.05022

openalex publication_date 2017/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let a_n be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order n^-α, 0 < α< 1/2. We prove that, almost surely, for every measure-preserving system (X,T) and every f ∈ L¹(X) orthogonal to the invariant factor, the modulated, random averages sup_b | (1)/(N) ∑_{n = 1}^N b(n) T^a_n f | converge to 0 pointwise almost everywhere, where the supremum is taken over a set of bounded functions with certain uniform approximation properties.

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