Limit Theorems for a class of unbounded observables with an application to "Sampling the Lindelöf hypothesis"

Abstract

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise $C^2$ expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on $\mathbb{R}$. The class of observables in the CLT and the MLCLT on $\mathbb{R}$ include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber (2009) and Steuding (2012) who have proven the Strong Law of Large Numbers for "Sampling the Lindelöf hypothesis".

Version as in the journal with an extension of Remark 2.16

References in corpus (5)

• Edgeworth expansions for weakly dependent random variables
• Sampling the Lindelöf Hypothesis with the Cauchy Random Walk
• Expansions in the local and the central limit theorems for dynamical systems
• An ergodic value distribution of certain meromorphic functions
• Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails