Lacunary Sums Demonstrate Central Limit Theorem and Law of Iterated Logarithm under Hadamard Gap Condition

The behaviour of lacunary sums, mathematical expressions involving sequences of dilated functions, closely mirrors that of independent random variables, but this connection weakens when trigonometric functions are replaced with more general periodic functions, introducing sensitivity to the underlying arithmetic properties of the sequence. Christoph Aistleitner, Lorenz Fruehwirth, and Joscha Prochno demonstrate a sharp threshold governing the tail probabilities of these sums, revealing a precise point at which their behaviour transitions from predictable, Gaussian patterns to potentially chaotic fluctuations. Their work establishes that the number of solutions to a specific Diophantine equation dictates this change, with sums behaving in accordance with standard normal distributions up to a certain threshold, beyond which this regularity breaks down. This research not only refines understanding of the link between arithmetic and probability, but also proves this threshold is optimal, representing a fundamental limit on the predictability of lacunary sum behaviour.

The probability space encompassing values between zero and one, with its standard mathematical properties, adheres to both the central limit theorem and the law of the iterated logarithm. However, the situation becomes considerably more complex when a simple trigonometric function is replaced by a more general periodic function, and the arithmetic properties of a sequence of numbers become crucial. Recent research demonstrates that the validity of the law of the iterated logarithm requires a stricter mathematical condition than the central limit theorem, extending these findings to establish a broader connection between these arithmetic properties and the rates of convergence in the functional central limit theorem.

Lacunary Trigonometric Series, Normal Convergence, Large Deviations

This research investigates the convergence of lacunary trigonometric series to a normal distribution, establishing principles for understanding rare events within these series. Lacunary series are trigonometric sums where the terms are spaced out irregularly, making their analysis more challenging than standard Fourier series. The study focuses on how well the sequence generated by these terms is distributed, measuring its evenness. The law of the iterated logarithm describes the long-term behavior of sums of independent random variables, while large deviation principles describe the probability of rare events.

The central limit theorem states that the sum of many independent random variables approaches a normal distribution. The primary goal of this work is to establish principles for understanding the probability of unusually large deviations in the distribution of lacunary sequences. The authors prove that the probability of these deviations can be estimated using exponential functions, connecting this deviation to the partial sums of the lacunary trigonometric series and refining existing results with more precise estimates. The paper follows a standard mathematical research format, beginning with an introduction that provides background and summarizes the main results.

It then defines the necessary concepts and auxiliary results before presenting the main theorems, followed by detailed mathematical proofs and illustrative examples. The paper concludes by summarizing the findings and suggesting directions for future research. The authors employ a variety of mathematical techniques, including Fourier analysis, probability theory, harmonic analysis, and the study of exponential sums. They also utilize techniques from number theory to analyze the distribution of the sequence, contributing to our understanding of lacunary trigonometric series and their applications in number theory, signal processing, and dynamical systems, and developing new tools for analyzing the convergence of random series.

Lacunary Sums Converge to Normal Distribution

Scientists have established a precise connection between the arithmetic properties of a sequence of numbers and the probabilistic behavior of associated mathematical sums. The research centers on lacunary systems, which are sums derived from sequences of integers with increasing gaps, and demonstrates how controlling the number of solutions to specific equations dictates the behavior of these sums. The team proved that if the number of solutions to a particular equation is limited, then the probability of the sum exceeding a certain threshold aligns with standard normal behavior up to a specific point. Specifically, the work shows that for any periodic function, the probability that the lacunary sum exceeds a certain value is asymptotically equivalent to the standard normal distribution function, provided that value remains below a certain limit.

This result establishes a clear boundary, demonstrating that the tail probabilities of the lacunary sums behave predictably as long as this limit is not exceeded. The researchers rigorously proved this criterion is optimal, meaning that exceeding this threshold leads to a dramatically different outcome. Experiments revealed that beyond this critical value, the probability of the sum exceeding the threshold diverges from the standard normal distribution, indicating erratic behavior, and that beyond this threshold, there exists a function for which the probability of the sum exceeding the threshold approaches infinity. This transition in behavior highlights the delicate interplay between the number of solutions and the probabilistic characteristics of the lacunary sums, providing a precise mathematical framework for understanding their behavior.

Gaussian Convergence Governed by Integer Solutions

Researchers have established a precise boundary defining when the behavior of sums of periodic functions transitions from predictable Gaussian patterns to potentially erratic fluctuations. The work centers on sequences of integers and their impact on the convergence of these sums, building upon earlier observations that these sequences often mimic the properties of independent random variables. Specifically, the team demonstrated that the number of solutions to a particular equation, relating the terms within the integer sequence, acts as a critical threshold. If the number of solutions to this equation remains below a certain level, the probabilities of deviations from the expected Gaussian behavior remain consistent with standard normal distributions. Importantly, the researchers proved this criterion is optimal, meaning that exceeding this threshold does indeed lead to a departure from Gaussian behavior, refining our understanding of how arithmetic properties of integer sequences influence the convergence of sums, extending beyond the well-established results for independent random variables. The authors acknowledge that determining the precise nature of the erratic behavior beyond this threshold remains a challenge for future investigation, and that further research could explore the characteristics of these non-Gaussian distributions and the conditions under which they arise, potentially revealing new insights into the interplay between number theory and probability.