Normal approximation for iterated inner functions

Abstract: A Berry--Esséen theorem for linear combinations of iterates of an inner function is obtained. Our proof, which is based an elementary transfer argument and classical results in martingale theory, also leads to a simple proof of Nicolau and Soler i Gibert's central limit theorem for inner functions.

• The paper establishes a Berry–Esseen theorem for linear combinations of iterated inner functions, deriving a quantitative convergence rate of O(N^{-1/4+ε}).
• The paper leverages transfer identities and reverse martingale differences to transform the problem into a form suitable for applying central limit theorem techniques.
• The paper's findings impact the study of holomorphic dynamics and ergodic theory, opening pathways for further applications in operator theory and randomized algorithms.

Berry–Esseen Bounds for Iterated Inner Functions and Their Central Limit Behavior

Introduction

The paper "Normal approximation for iterated inner functions" (2604.16002) establishes a quantitative Berry–Esseen theorem for linear combinations of iterates of non-rotational inner functions on the unit disk. By leveraging transfer identities and reverse martingale difference techniques, it strengthens and simplifies recent central limit theorems (CLT) for such systems. This essay provides a comprehensive overview of the mathematical contributions, structure, technical machinery, and broader implications of the results.

Inner Functions, Iteration, and Measure-Theoretic Dynamics

Inner functions—analytic self-maps of the open unit disk $\mathbb{D}$ whose boundary values have modulus one almost everywhere—are fundamental objects in complex analysis and functional analysis. When such a function $f$ satisfies $f(0) = 0$, it preserves the normalized Lebesgue measure $m$ on the unit circle $\mathbb{T}$. The iteration of these maps, $f^{\circ n}$, yields nontrivial measure-preserving dynamics with deep links to ergodic theory and operator theory.

Understanding the statistical distribution of

$S_N = \sum_{n=1}^N a_n f^{\circ n}$

where $(a_n)$ is a complex coefficient sequence, is a central structural question both for spectral analysis and for applications involving randomization and stability in holomorphic dynamics.

Main Results: Berry–Esseen Theorem and Quantitative CLT

The principal outcome is a Berry–Esseen–type theorem that quantifies the convergence rate of properly normalized linear combinations of iterated inner functions to a complex normal law. Explicitly, for $f$ an inner function with $f(0)=0$ and not a rotation, and $f$0 satisfying

$f$1

setting

$f$2

the supremum deviation of the real projection $f$3 from the standard normal is bounded by

$f$4

where the explicit decay rate is determined by fourth moments and the Lindeberg-type conditions, and $f$5 is the key derivative parameter. In the classical case $f$6, the upper bound is $f$7 for arbitrary $f$8.

This sharp rate is a significant refinement over the nonquantitative central limit result:

$f$9

previously proved by Nicolau and Soler i Gibert, by yielding a quantitative approximation and thus establishing the speed of statistical stabilization in such dynamical systems.

Proof Strategy: Reverse Martingale Differences and Transfer Arguments

The innovation of the approach lies in a transfer argument which recasts sums of iterated inner functions into sums of reverse martingale differences, plus a negligible boundary term. Explicitly, using the coefficients

$f(0) = 0$0

there is an identity:

$f(0) = 0$1

with martingale properties for $f(0) = 0$2, analogous to classical orthogonality and difference sequence controls in stochastic processes.

The crucial technical step then applies the martingale CLT (Brown–Eagleson, Haeusler) to this decomposition. Detailed control of variances, higher moments, and maximal terms ensures the Lindeberg and uniform integrability conditions are enforced and thus yield normal approximation for the sums. Importantly, the paper derives uniform Berry–Esseen–type error bounds across all unit circle directions, using precise moment calculations and martingale inequalities.

Numerical and Asymptotic Rates

A strong numerical claim of the paper is the $f(0) = 0$3 bound for the Kolmogorov distance between the distribution of the normalized sum and the limiting complex normal, in the case of constant coefficients.

This rate provides a quantitative answer to questions about decorrelation and statistical mixing in holomorphic dynamical systems, advancing beyond previous qualitative measures and opening analysis of finer-scale behavior.

Theoretical Implications and Connections

The results strengthen the interface between complex dynamics, ergodic theory, and stochastic process theory. By providing quantitative normal approximations for noncommutative systems arising from analytic function iterations, the methodology connects spectral theory for composition operators, martingale techniques, and quantitative central limit theory.

From a practical perspective, the rates are relevant for numerically estimating statistical quantities in holomorphic dynamics, high-dimensional signal processing, and invariance principles for randomized algorithms built from analytic or measure-preserving maps.

On the theoretical side, the transfer reduction potentially extends to broader classes of noncommutative dynamical models, suggesting possible generalizations to other functional equations, operator-theoretic contexts, and beyond holomorphic setting.

Future Directions

Open problems suggested by the work include:

• Determining the optimal rate decay for the Berry–Esseen bounds in this setting.
• Extending this approach to more general (non-inner) analytic or quasiconformal maps.
• Analyzing functional versions of the CLT for vector-valued or operator-valued iterates.
• Applying these techniques in ergodic-theoretic rigidity settings, where the presence of additional algebraic or geometric structure may yield new results.

Conclusion

This paper provides a rigorous and quantitative advancement in the study of statistical properties of iterated inner functions, establishing explicit normal approximation rates using martingale transfer and Berry–Esseen techniques. Its methodology not only clarifies and extends recent results in the field, but also sets a standard for further quantitative analyses of limit theorems in holomorphic and dynamical frameworks.