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$L^p$ maximal inequalities for Gaussian subordination with applications to Breuer--Major--Donsker principles

This paper establishes a new $L^2$ maximal inequality for partial sums of Gaussian-subordinated random variables that depends only on the Hermite rank and covariance structure, thereby proving the Breuer--Major--Donsker principle under the sole finite-variance assumption and removing previously required stronger integrability or structural conditions.

Original authors: Dionysis Milesis, Guangqu Zheng

Published 2026-10-08

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Dionysis Milesis, Guangqu Zheng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of randomness, scientists often look for patterns in noise. Imagine a long line of measurements taken over time, where each new number is influenced by the ones before it. This is the world of dependent random variables, a common feature in everything from weather patterns to financial markets. When these numbers are generated by a Gaussian process—a specific, bell-curve-shaped distribution of chance—mathematicians have long sought to understand how their sums behave as the list grows longer. A central question is whether these sums, when scaled down, settle into a predictable, smooth curve known as Brownian motion, the mathematical model for a particle drifting in a fluid. For decades, proving this convergence required strict rules about how "wild" the individual numbers could be, often demanding that extreme outliers be mathematically impossible or exceedingly rare.

A team of researchers has now removed these strict rules, showing that the smooth, predictable behavior emerges even when the individual numbers are allowed to be much more erratic, provided only that their average size is finite. They established a new mathematical guardrail, an inequality, that controls the maximum distance a sum can wander from its starting point. This result is significant because it proves that the familiar laws of large numbers and the central limit theorem hold true under the weakest possible condition: that the variance, or the average squared distance from the mean, is finite. By doing so, they have unified the understanding of these random processes, showing that the complex machinery of dependence does not require extra safety nets to produce the same elegant, universal limits seen in independent systems.

The researchers focused on a specific type of random variable created by applying a function to a Gaussian process. Think of this as taking a standard, bell-shaped curve of chance and running it through a machine that transforms the output. If the machine is simple, the result is straightforward; if the machine is complex, the result can be highly dependent on the input. The team wanted to know how the sum of these transformed numbers behaves over time. Previous work had shown that if the transformed numbers were well-behaved—meaning they didn't have heavy tails or extreme spikes—the sums would converge to a smooth path. However, those proofs relied on assumptions that the numbers had to be even more well-behaved than just having a finite average size. The new work demonstrates that this extra layer of safety is unnecessary.

To achieve this, the authors developed a new method for bounding the maximum value a partial sum can reach. In probability, a "maximal inequality" is a tool that limits how far a running total can deviate from zero at any point in time. For independent random numbers, this limit is well understood. For dependent numbers, the path is much more treacherous because a large value at one step can influence the next. The researchers found that by carefully analyzing the structure of the dependence, they could prove a limit that scales with the square root of the number of steps, which is the natural scale for these processes. Crucially, they managed to do this without the "logarithmic loss" that had plagued previous attempts at the critical boundary of finite variance. In simpler terms, earlier methods would produce a bound that was slightly too loose, growing a bit faster than the true limit, which prevented a clean proof. The new method tightens this bound to the exact optimal scale.

The proof involves a clever strategy of breaking the problem into manageable pieces. The researchers first truncated the function, effectively cutting off the most extreme values to create a simpler, bounded version of the problem. They then analyzed the "Hermite rank," a measure of how complex the relationship between the input and output is, to determine the strength of the dependence. By separating the problem into low-order and high-order components, they could apply different mathematical tools to each. A key innovation was a lemma that looked at two adjacent blocks of data simultaneously. Instead of treating each block in isolation, which led to the loose bounds of the past, looking at them together revealed a subtle cancellation effect that tightened the estimate. This allowed them to sum up the errors across different time scales without accumulating the extra logarithmic factor that had previously blocked the proof.

The implications of this finding extend to both discrete and continuous time. In the discrete setting, where data comes in a sequence of steps, the result confirms that the Breuer-Major-Donsker principle holds under the sole assumption of finite variance. This principle describes how a random walk converges to Brownian motion. Previously, proving this required additional assumptions about the integrability of the data or specific prediction-theoretic properties that were not always guaranteed. The new work removes these hurdles, showing that the convergence happens as long as the variance is finite. Similarly, for continuous-time processes, where data flows like a stream rather than a sequence of steps, the researchers proved that the same principle applies without needing the extra moment assumptions that were previously thought necessary.

The team also applied their findings to self-similar processes, which are systems that look statistically similar at different scales, like the jagged edge of a coastline or the fluctuations of a stock price. In a specific "critical" regime where the dependence is just strong enough to change the behavior of the system, the researchers showed that the sums still converge, but they require a different normalization involving a logarithmic factor. They identified the exact variance of this limit, correcting a previous calculation in the literature. This result is particularly important because it handles the most delicate case where the system is on the verge of changing its fundamental behavior. By isolating the leading component of the randomness and controlling the rest, they demonstrated that the limit exists and is well-defined.

This work represents a refinement of our understanding of how randomness aggregates. It shows that the universe of dependent random variables is more robust than previously thought. The requirement for the sums to behave predictably is not as fragile as once believed; it does not crumble under the weight of heavy tails or complex dependence structures, provided the average size of the fluctuations is finite. The researchers have provided a rigorous mathematical foundation that removes artificial constraints, allowing the natural laws of probability to shine through in their purest form. Their methods, which combine truncation, diagrammatic analysis, and a novel approach to adjacent blocks, offer a new toolkit for tackling similar problems in stochastic processes.

The results are not merely theoretical curiosities; they clarify the conditions under which standard statistical models can be trusted. In fields ranging from physics to finance, where data often exhibits long-range dependence, knowing that the central limit theorem holds under minimal assumptions provides a stronger basis for inference. The paper does not claim to solve every problem in this field, nor does it suggest that all dependent systems behave identically. Rather, it establishes a precise boundary: finite variance is sufficient. Anything less, and the behavior may be unpredictable; anything more, and the result holds. This clarity allows scientists to apply these principles with greater confidence, knowing exactly what conditions are required for the smooth, Brownian motion to emerge from the chaos of dependent data.

Technical Summary: $L^p$ Maximal Inequalities for Gaussian Subordination with Applications to Breuer–Major–Donsker Principles

Problem Statement
This paper addresses the problem of establishing sharp maximal inequalities for partial sums of dependent random variables obtained via Gaussian subordination, specifically of the form $X_i = f(G_i)$ , where $(G_i)$ is a jointly Gaussian family and $f$ is a measurable function. The central challenge lies in the critical case $p=2$ . While maximal inequalities for independent variables or martingales are well-understood (yielding an optimal $\sqrt{n}$ scale), dependent sums typically incur a logarithmic loss in the $L^2$ norm when using standard dyadic arguments. Previous results for Gaussian subordination required stronger assumptions, such as $f \in L^p(\gamma)$ for some $p > 2$ , or specific prediction-theoretic conditions, to establish functional limit theorems (tightness) under the finite-variance assumption ( $f \in L^2(\gamma)$ ). The paper seeks to remove these extra integrability and structural assumptions, proving that the optimal $\sqrt{n}$ scale holds for $f \in L^2(\gamma)$ provided the covariance structure satisfies a uniform row-sum bound determined by the Hermite rank of $f$ .

Methodology
The authors employ a multi-step strategy combining probabilistic truncation, chaos decomposition, and combinatorial estimates:

Rank-Preserving Truncation: To handle the unbounded nature of general $L^2$ functions, the authors truncate $f$ at level $\sqrt{n}$ and subtract the low-order Hermite projections created by this truncation. This yields a function $g_n$ that retains the original Hermite rank $d$ but has controlled higher moments ( $L^4$ ).
Low–High Chaos Decomposition: The square of the truncated function, $g_n^2$ , is decomposed into low-order Hermite chaoses (orders $1$ to $d-1$ ) and a high-order remainder. The low-order terms are controlled using Gaussian small-set estimates and logarithmic bounds, while the high-order part is handled via orthogonality and hypercontractivity.
Separated Diagram Estimates: The proof utilizes refined estimates for products of Hermite polynomials evaluated at distinct Gaussian coordinates. These estimates, inspired by Bardet and Surgailis, bound the moments of sums by exploiting the decay of covariance powers. A key innovation is a "mixed-rank" refinement that improves bounds when one factor belongs to a lower chaos.
Adjacent-Block Maximal Lemma: A critical technical component is a lemma that converts a strict power gain in the fourth-moment estimate of two adjacent disjoint blocks into a maximal second-moment bound. Unlike standard dyadic arguments that sum over independent scales, this approach couples adjacent blocks to recover the $\sqrt{n}$ scale without logarithmic loss.
Finite Partition and Recombination: To handle general covariance structures where coordinates may be strongly correlated, the authors partition the index set into a finite number of classes (depending only on the Hermite rank $d$ and the row bound $K$ ) such that coordinates within each class have uniformly small correlations. The maximal inequality is applied to each class and recombined.
Extension to $p > 2$ : For $p > 2$ , the authors utilize the Malliavin–Meyer argument (Nourdin and Nualart) combined with dyadic summation. Here, the geometric decay of the dyadic series ensures convergence, making the $p=2$ case the critical endpoint.

Key Contributions and Results

$L^2$ Maximal Inequality (Theorem 1.1): The paper establishes that for a function $f \in L^2(\gamma)$ with Hermite rank at least $d$ , and a Gaussian family satisfying a uniform $d$ -th power row bound $K$ , the maximal partial sum satisfies:
$\mathbb{E}\left[ \max_{0 \le m \le n} \left| \sum_{k=1}^m f(Z_k) \right|^2 \right] \le C_{d,K} n \|f\|_{L^2(\gamma)}^2$
This result is uniform in $n$ and the covariance matrix, removing the logarithmic factor $\log n$ typically associated with dyadic arguments at $p=2$ .
Stationary Breuer–Major–Donsker (BMD) Principles: As a direct consequence, the paper proves the functional central limit theorem (tightness and convergence in distribution) for stationary Gaussian subordinated processes under the sole assumption of finite variance ( $f \in L^2(\gamma)$ ) and the summability of the $d$ -th power of the covariance function ( $\sum |\rho(k)|^d < \infty$ ). This removes the additional $L^{2+\epsilon}$ moment assumptions found in Nourdin and Nualart (2020) and the prediction-theoretic assumptions in Mansanarez, Poly, and Zheng (2026).
Continuous-Time and Self-Similar Extensions: The results are extended to continuous-time stationary processes and non-stationary self-similar Gaussian processes (following Campese, Nourdin, and Nualart). In the self-similar setting, the authors remove the extra moment assumptions for the subcritical regime ( $\alpha < 2 - 1/d$ ) and provide a logarithmically normalized limit for the critical regime ( $\alpha = 2 - 1/d$ ).

Significance
The paper claims that its primary contribution is the resolution of the critical $p=2$ endpoint for maximal inequalities in Gaussian subordination. By eliminating the logarithmic loss, the authors demonstrate that the finite-variance assumption is sufficient for the Breuer–Major–Donsker principle, unifying the conditions for finite-dimensional convergence and functional convergence (tightness). This removes the need for stronger integrability conditions or specific structural assumptions (like decimation or prediction errors) that were previously required to establish tightness. The work thus provides a more robust and general framework for limit theorems of dependent Gaussian sequences and processes, relying only on the natural finite-variance condition and the covariance decay dictated by the Hermite rank.

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