[Paper Review] Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type

Marc Keßeböhmer, Tanja I. Schindler|arXiv (Cornell University)|Jan 13, 2019

Mathematical Dynamics and Fractals34 references7 citations

TL;DR

This paper establishes intermediate trimming strong laws of large numbers for Birkhoff sums on subshifts of finite type by introducing a new Banach space of quasi-Hölder continuous functions, which enables spectral gap analysis and extends prior results from interval maps to symbolic dynamics. The key contribution is proving that for observables with regularly varying tails (e.g., St. Petersburg-type distributions), an intermediately trimmed sum normalized by a suitable sequence converges almost surely to 1, even when the observable is non-integrable.

ABSTRACT

We prove strong laws of large numbers under intermediate trimming for Birkhoff sums over subshifts of finite type. This gives another application of a previous trimming result only proven for interval maps. In case of Markov measures we give a further example of St.\ Petersburg type distribution functions. To prove these statements we introduce the space of quasi-H\"older continuous functions for subshifts of finite type.

Motivation & Objective

• To extend intermediate trimming strong laws of large numbers from piecewise expanding interval maps to subshifts of finite type.
• To overcome the failure of standard Lipschitz functions in satisfying spectral gap and truncation conditions in the dynamical systems setting.
• To introduce and analyze the space of quasi-Hölder continuous functions as a larger, more suitable Banach space for spectral theory on subshifts.
• To establish new limit theorems for truncated sums and provide a refined analysis of St. Petersburg-type distributions in the context of Gibbs-Markov measures.
• To demonstrate that the same trimming and normalization sequences work for both i.i.d. sequences and a large class of dynamical systems with regularly varying tail distributions.

Proposed method

• Introduce a new Banach space of quasi-Hölder continuous functions on subshifts of finite type, defined via a norm involving oscillation over shrinking balls with respect to a measure-based metric.
• Prove that this space forms a Banach algebra and supports a spectral gap for the transfer operator associated with the shift map.
• Establish that the transfer operator on this space satisfies the spectral gap property under suitable conditions on the potential function.
• Use the spectral gap to verify the conditions of Property D from prior work (KS18), enabling the application of intermediate trimming strong laws.
• Define intermediate trimming by removing the largest bn summands from Birkhoff sums, with bn = o(n), and derive almost sure convergence to 1 after normalization.
• Apply the theory to Markov measures and construct explicit examples of St. Petersburg-type distributions on subshifts, showing convergence with the same trimming and normalization as in the i.i.d. case.

Experimental results

Research questions

• RQ1 Can intermediate trimming strong laws be extended from interval maps to subshifts of finite type, where standard Lipschitz functions fail to satisfy required spectral and truncation conditions?
• RQ2 Does the space of quasi-Hölder continuous functions provide a suitable framework for spectral gap analysis in symbolic dynamics, particularly for non-integrable observables?
• RQ3 For observables with regularly varying tails (e.g., α ∈(−1,0)), does the same trimming sequence bn and normalization dn yield almost sure convergence in subshifts as in the i.i.d. case?
• RQ4 Can the theory of truncated sums be improved for non-integrable random variables in dynamical systems, particularly in the context of Gibbs-Markov measures?
• RQ5 Do St. Petersburg-type distributions on subshifts of finite type exhibit the same asymptotic behavior as in the i.i.d. setting under intermediate trimming?

Key findings

• The space of quasi-Hölder continuous functions forms a Banach algebra with a norm that supports a spectral gap for the transfer operator on subshifts of finite type.
• The transfer operator on this space satisfies the spectral gap property, enabling the application of intermediate trimming results from prior work (KS18).
• For observables with regularly varying tails (α ∈(−1,0)), an intermediate trimming strong law holds with the same bn and dn sequences as in the i.i.d. case.
• The paper constructs explicit examples of St. Petersburg-type distributions on subshifts with Gibbs-Markov measures, demonstrating that the same normalization applies.
• The truncated sum Tfn_n χ converges almost surely under the same conditions as in the i.i.d. case, improving on prior results by relaxing truncation sequence conditions.
• The compactness of the unit ball in the quasi-Hölder norm is proven via approximation by cylinder functions, ensuring convergence of subsequences and supporting spectral theory.