Zobrazit v EDS

Fluctuation of ergodic averages and other stochastic processes.

Uloženo v:
Podrobná bibliografie
Název: Fluctuation of ergodic averages and other stochastic processes.
Autoři: Mondal, S., Rosenblatt, J., Wierdl, M.
Zdroj: Transactions of the American Mathematical Society; Sep2025, Vol. 378 Issue 9, p6453-6484, 32p
Témata: ERGODIC theory, STOCHASTIC processes, UNIFORM distribution (Probability theory), MARTINGALES (Mathematics), OSCILLATIONS, INTEGRABLE functions, STOCHASTIC convergence, MATHEMATICAL functions
Abstrakt: For an ergodic map T and a non-constant, real-valued f\in L^1, the ergodic averages A_Nf(x) =\frac 1N\sum \limits _{n=1}^N f (T^n x) converge a.e., but the convergence is never monotone. Depending on particular properties of the function f, the averages A_Nf(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\in L^1 has the averages A_Nf(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. [ABSTRACT FROM AUTHOR]
Copyright of Transactions of the American Mathematical Society is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Databáze: Complementary Index
Popis
Abstrakt:For an ergodic map T and a non-constant, real-valued f\in L^1, the ergodic averages A_Nf(x) =\frac 1N\sum \limits _{n=1}^N f (T^n x) converge a.e., but the convergence is never monotone. Depending on particular properties of the function f, the averages A_Nf(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\in L^1 has the averages A_Nf(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. [ABSTRACT FROM AUTHOR]
ISSN:00029947
DOI:10.1090/tran/9429