Trimmed Ergodic Sums in Dynamical Systems

• Trimmed ergodic sums are defined as partial sums along orbits that systematically remove the largest observations to mitigate heavy-tailed effects.
• This methodology restores classical limit behaviors by eliminating rare, dominating events that thwart standard ergodic theorems.
• Applications include chaotic maps and irrational rotations, linking extreme-value statistics with system-specific mixing properties and Diophantine phenomena.

Trimmed ergodic sums are partial sums of observables evaluated along orbits of a dynamical system in which the largest (often pathological) values are systematically removed. This methodology restores classical limit behavior—such as (strong) laws of large numbers—for Birkhoff sums arising from non-integrable observables, where standard ergodic theory fails due to heavy tails or power-law singularities. Trimmed sums have become fundamental in the analysis of dynamical systems with non-integrable observables, with their behavior tightly controlled by system-specific mixing properties and the extremal statistics of the observable.

1. Definitions and Core Framework

Let $(X, \mathcal{B}, \mu, T)$ be an ergodic, measure-preserving dynamical system, and let $f: X \to [0, \infty)$ be a nonnegative observable. The classical Birkhoff sum is

$S_n(f,x) = \sum_{k=0}^{n-1} f(T^k x).$

For observables with non-integrable or heavy-tailed distributions, the sums can be dominated by a vanishingly small number of exceptionally large terms, so that no normalization yields almost sure convergence. The trimmed sum, by contrast, removes the $k_n$ largest observations: $$S_n^{(k_n)}(f)(x) = \sum_{j=0}^{n-1} f(T^j x) - \sum_{i=1}^{k_n} f(x_i^n),$$ where $f(x_1^n) \geq \ldots \geq f(x_n^n)$ are the ordered values in the orbit segment $[x, T^{n-1}x]$.

Two principal regimes are considered:

• Light trimming: $k_n = K$ fixed, typically $K = 1$.
• Intermediate trimming: $k_n \to \infty$, $k_n = o(n)$ as $n \to \infty$.

The specific choice of $k_n$ is dictated by the tail properties of $f$ and the dynamics of $T$.

2. Motivation: Failure of Classical Laws and the Role of Trimming

For $f \notin L^1(\mu)$, Aaronson's theorem establishes that the standard strong law cannot hold: there is no nontrivial normalization $a_n \to \infty$ such that $S_n(f, x) / a_n \to 1$ almost surely. Instead, rare but overwhelmingly large summands—arising, for example, from returns near singularities—dominate the sums. Removing these via trimming reveals the regular bulk behavior masked by the extremes. The efficacy and necessity of trimming depend on the interplay between orbit structure, observable tail, and statistical independence or mixing of the system (Schindler, 2018, Auer et al., 28 Mar 2025).

In mixing systems, the spatial and temporal distribution of extremes often mimics the i.i.d. case, and trimming a vanishing fraction suffices. In rank-one or minimal-mixing systems, such as irrational rotations, clustering of extremes enforces Diophantine constraints on the rotation angle for any trimmed law to hold (Auer et al., 28 Mar 2025).

3. Key Results: Laws for Trimmed Sums Across Dynamical Systems

3.1. Exponentially Mixing and Chaotic Systems

For $f$ with a power singularity at a point $x^*$, such that $\mu\{f > t\} \asymp t^{-1/\alpha}$, the regime is controlled by $\alpha = \beta / \dim M$. In exponentially mixing dynamical systems, the following hold (Auer et al., 13 Jan 2026):

• Strong law ($\alpha \ge 1$):
  • With $k_n \to \infty$, $k_n = o(n)$ and moderate growth,

  $$\frac{S_n^{(k_n)}(f)}{n^\alpha k_n^{1-\alpha}} \to \frac{\operatorname{Res}_{x^*}(f) (B_d \rho(x^*))^\alpha}{\alpha-1}$$

  almost surely, where $B_d$ is the unit ball in $\mathbb{R}^d$ and $\rho(x^*)$ is the density at the singularity.

• Light trimming and non-standard limit theorems ($1/2 < \alpha \leq 1$):

  • Centered and normalized lightly trimmed sums converge in distribution to an explicit, non-standard law involving a Poisson process (see Section 4 for details).

3.2. Non-mixing Systems: Irrational Rotations

For $T_\alpha: x \mapsto x + \alpha \pmod{1}$, and $f(x) = x^{-\beta}$ ($\beta \geq 1$), the following precise regime distinctions are established (Auer et al., 28 Mar 2025):

• Weak law ($\beta = 1$): For Roth-type $\alpha$, no trimming yields

$\frac{S_n(f,x)}{n \log n} \to 1$

in probability as $n \to \infty$. If $\alpha$ is not Roth-type, even light trimming fails, and only near-complete trimming can restore convergence.

• Strong law ($\beta = 1$, bounded type $\alpha$): The single-term trimmed sum

$\frac{S_n^{(1)}(f,x)}{n \log n} \to 1$

for almost all $x$.

• Strong law ($\beta > 1$): For growing but vanishing trimming $r_n \to \infty$, $r_n = o(n)$,

$$\frac{S_n^{(r_n)}(f,x)}{n^{1-\beta}/(\beta-1)} \to 1.$$

4. Quantitative Rates and Error Analysis

Trimmed ergodic sum theorems often admit effective quantitative rates. For weakly integrable or heavy-tailed observables in uniformly mixing systems (Haynes, 2011):

• The leading term in the trimmed sum is $N \cdot F_1(N)$ (the truncated mean), with error $E_N = F_3(N)^{2/3} \log^{1/3+\epsilon} F_3(N)$, where $F_3(N)$ aggregates contributions from second moments and mixing rates.
• In the Gauss map and continued-fraction context, this produces explicit expansions with secondary and error terms for the digit sums, sharpened via a single-term trimming device:

$$\sum_{n=1}^N a_n(x) - \max_{1\le n \le N} a_n(x) = N \log N + O_\epsilon(N^{2/3} \log^{5/3 + \epsilon} N)$$

Such results extend trimmed sum theory considerably beyond the purely i.i.d. setting.

5. Diophantine Phenomena and the Maximal Summand

Recent work (Kanigowski et al., 25 Aug 2025) has elucidated a sharp Diophantine criterion for the so-called "extravagance" of the maximal summand in Birkhoff sums for observables with multiple non-integrable singularities (e.g., $\phi(x) = x^{-1} - (1-x)^{-1}$), under irrational rotation. Writing

$$\limsup_{N \to \infty} \frac{\max_{0\le k < N} \phi(R_\alpha^k(x))}{S_N(\phi)(x)}$$

they show that this limsup is either $0$ or $\infty$, depending on the convergence or divergence of the series $\sum_n W_n(\alpha)$, with $W_n(\alpha)$ computed from the continued-fraction entries of $\alpha$. The exceptional zero set has Hausdorff dimension $1/2$. This establishes that the extreme-value phenomenology of trimmed sums is intricately linked to fine arithmetic properties of the transformation.

6. Infinite Measure, Induced Systems, and Applications

In infinite-measure preserving transformations, where $\mu(X) = \infty$, classical laws fail even more spectacularly. Trimmed sums and the addition of a random number of extra terms both recover strong laws, and are, in a specific technical sense, equivalent (Bonanno et al., 2023):

• If one trims a fixed number $r$ (typically one) of the largest values per induced block, the sum, normalized appropriately, converges to a nontrivial limit.
• If the observable has "flattened tails," a single excursion removal suffices; for heavier tails, a growing number of extreme values must be trimmed.
• Application to non-regular continued-fraction algorithms (e.g., Rényi-type, even-integer) yield explicit limit results for digit sum statistics.

7. Probabilistic Model Systems and Classical Expansions

In random models such as generalized Oppenheim expansions and associated Lüroth, Engel, and Sylvester series, strong laws for lightly trimmed sums have been established using i.i.d.-type arguments when possible (Hadjikyriakou et al., 2023):

• For i.i.d.-type tail structure, trimming the single largest value restores a classical normalization $n \log n$ with a non-zero limit constant.
• In broader, non-i.i.d. contexts, only a very weak (vanishing) law can be proved with the same normalization.
• This succinctly demonstrates the necessity of both system-specific ergodic structure and observable tail control for strong trimmed sum laws.

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Summary Table: Typical Trimmed SLLN Regimes

System type	Observable tail	Trimming regime	Normalization	SLLN lim	Source
Exp. mixing (e.g., chaotic)	$\mu(f>t) \sim t^{-1/\alpha}$	$k_n \to \infty$, $k_n = o(n)$ or $k_n=K$	$n^\alpha k_n^{1-\alpha}$ (interm.), $n\log n$ (light)	Non-trivial const.	(Auer et al., 13 Jan 2026)
Irrational rotation (bounded-type $\alpha$)	$f(x)=x^{-\beta}, \beta\geq 1$	$r_n=1$ ($\beta=1$), $r_n\to\infty$ ($\beta>1$)	$n\log n$ ($\beta=1$), $n^{1-\beta}/(\beta-1)$ ($\beta>1$)	1	(Auer et al., 28 Mar 2025)
Oppenheim, Lüroth expansions	i.i.d. type tails	$r=1$ (light)	$n\log n$	$\alpha>0$	(Hadjikyriakou et al., 2023)
Infinite measure, induced map	As in (ii)	$r=1$, $r\to\infty$	$N$	const.	(Bonanno et al., 2023)

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Trimmed ergodic sums reveal that rare, massive deviations in heavy-tailed ergodic processes can be systematically controlled, providing a unified mechanism for restoring strong and weak limit theorems across a diverse array of dynamical regimes. Their study has catalyzed a new interface between ergodic theory, extreme-value statistics, arithmetic, and fractal phenomena.