Fluctuation of ergodic averages and other stochastic processes

2024/04/17 by Sovanlal Mondal, Mondal, Sovanlal, Joe Rosenblatt +3
Economics, Econometrics and Finance · #37A05 (Secondary) #37A46 #37A50 (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2404.11507

openalex publication_date 2024/04/17 · openalex created_date 2024/04/19 · openalex updated_date 2026/07/28

Abstract

For an ergodic map T and a non-constant, real-valued f ∈ L¹, the ergodic averages \mathbbA_N f(x) = \frac1 N ∑_n=1^N f(Tⁿ x) converge a.e., but the convergence is never monotone. Depending on particular properties of the function f, the averages \mathbbA_N f(x) may or may not actually fluctuate around the mean value infinitely often a.e. We will prove that a.e. fluctuation around the mean is the generic behavior. That is, for a fixed ergodic T, the generic non-constant f∈ L¹ has the averages \mathbbA_N f(x) fluctuating around the mean infinitely often for almost every x. We also consider fluctuation for other stochastic processes like subsequences of the ergodic averages, convolution operators, weighted averages, uniform distribution and martingales. We will show that in general, in these settings fluctuation around the limit infinitely often persists as the generic behavior.

Related

• Moving Averages
• Quantitative Convergence for Sparse Ergodic Averages in L¹
• Randomly Perturbed Ergodic Averages
• Absolute and Unconditional Convergence of Series of Ergodic Averages and Lebesgue Derivatives
• Rates of Divergence of non-Conventional Ergodic Averages