Distribution of random multiplicative functions in short intervals, with proper normalization

Abstract: We determine the limiting distribution of partial sums of a Steinhaus random multiplicative function $\sum_{x\le n \le x+y} f(n)$ over short intervals $[x, x+y]$, where $y \rightarrow \infty$ but $y=o(x)$. We show that with appropriate normalization, the limiting distribution is Gaussian for all such $y$. A key new feature of our result is that the normalization factor is different from the standard deviation $\sqrt{y}$ when $y$ is very close to $x$. In contrast, when $y \asymp x$ there is no normalization for which the limiting distribution is a non-degenerate Gaussian.

• The paper establishes the limiting distribution for S₍f₎(x, y) in short intervals by showing that a deterministic normalization leads to a Gaussian limit.
• It employs a martingale CLT with conditioning on small primes and barrier events to control contributions and ensure variance concentration.
• The results provide precise normalization techniques for short multiplicative sums, contrasting with the heavy-tailed, non-Gaussian behavior observed in long intervals.

Limiting Distribution of Random Multiplicative Functions in Short Intervals

Introduction and Problem Formulation

The paper "Distribution of random multiplicative functions in short intervals, with proper normalization" (2606.29040) addresses the distributional behavior of partial sums of Steinhaus random multiplicative functions in short intervals. Specifically, it considers the sum

$S_{f}(x, y) = \sum_{x < n \leq x+y} f(n)$

where $f(n)$ is a Steinhaus random multiplicative function—i.e., $f(p)$, for primes $p$, are independent random variables uniformly distributed on the unit circle, and $f(n)$ is extended multiplicatively to all $n$.

The primary goal is to establish the limiting distribution for $S_{f}(x, y)$ as $x \to \infty$ and $y \to \infty$ with $y = o(x)$—that is, in all genuinely short-interval regimes—under appropriate normalization, and to characterize the precise normalization required for asymptotic Gaussianity.

Background and Key Prior Results

Prior work—most notably by Chatterjee and Soundararajan [CS], Soundararajan and Xu [SoundXu], and Caich [caichshort]—had established Central Limit Theorems (CLTs) for $f(n)$0 when $f(n)$1 is relatively small compared to $f(n)$2, for example $f(n)$3 for some $f(n)$4. In these regimes, the standard normalization by $f(n)$5 yields a limiting (real or complex) Gaussian distribution:

$f(n)$6

However, as $f(n)$7 increases and approaches $f(n)$8, Harper's theorem on "better than square-root cancellation" [HarperLow] and subsequent analysis reveal this normalization becomes defective: the variance of long partial sums behaves asymptotically as $f(n)$9 rather than $f(p)$0, and no normalization yields a non-trivial Gaussian limit for full intervals.

Thus, for intervals $f(p)$1, the critical question is: what is the precise normalization for $f(p)$2 that yields a non-degenerate Gaussian limit, and how does this transition occur as $f(p)$3 approaches $f(p)$4?

Main Results and Normalization

The authors establish that for all $f(p)$5 with $f(p)$6, there exists a deterministic normalization $f(p)$7 such that

$f(p)$8

where the normalization is given by

$f(p)$9

More precisely, the scaling factor is explicitly described as

$p$0

This result is new in identifying the correct scaling even when $p$1 is very close to $p$2, revealing that the variance drops below $p$3 as $p$4 increases and approaches $p$5 (i.e., variance collapse).

A central technical feature is that this normalization is not simply the variance of $p$6 computed in the physical space but emerges as the asymptotic deterministic limit after conditional variance analysis and barrier events suppressing outlier behavior on the Fourier side.

Strong non-Gaussianity for long intervals is proven: for $p$7, no normalization yields Gaussian behavior—the limiting distribution is heavy-tailed, in line with predictions from multiplicative chaos [HarperLow, GW-Final].

Technical Approach

Martingale CLT with Conditionalization

The proof framework is based on a martingale central limit theorem a la McLeish [McLeish], with key refinements from [SoundXu]. Direct unconditioned approaches are stymied by failure of moment conditions for large $p$8. To break this barrier, the authors:

• Condition on small primes (all $p$9 for $f(n)$0).
• Show that, conditionally, after pruning the $f(n)$1-smooth part, the sum over rough numbers becomes amenable to the martingale CLT.

The second-moment and, crucially, the fourth-moment computations are then handled by more refined methods that combine Euler product Fourier techniques and barrier events, controlling the contribution from large deviations on the Fourier side.

Barrier Techniques and Concentration

A novel aspect is the use of barrier events to force the random Euler products (conditional variances) to stay within a high-probability corridor, suppressing the impact of heavy tails and rare configuration explosions. This analysis—grounded in probabilistic techniques from multiplicative chaos—yields sharp concentration of the conditional variance around $f(n)$2. Key ideas include:

• Reduction to Fourier side: Key statistics are analyzed via Parseval/Plancherel identities, bringing the problem to ergodic averages of random Euler products.
• Local independence: Exploited on the Fourier side to grant law of large numbers concentration once the effective "length" $f(n)$3 is large.
• Phase transition: As $f(n)$4 increases, concentration breaks for $f(n)$5, and Gaussianity is lost—connecting to phenomena in random matrix theory and critical multipicative chaos [HarperLow, GW-Final].

Fourth Moment and Off-Diagonal Control

For the auxiliary terms required in the martingale central limit theorem, off-diagonal contributions in conditional fourth moments are controlled by precise decorrelation estimates for random Euler products, relying on modern Gaussian process tools and explicit prime sum calculations.

Strong and Contradictory Claims

• Uniform Gaussianity holds for all $f(n)$6, $f(n)$7: This unifies and extends all prior results for short intervals by specifying the exact normalization even up to the threshold regime where $f(n)$8 is very close to $f(n)$9.
• No normalization yields a non-degenerate Gaussian for long sums: This is made quantitative via heavy-tail estimates; for $n$0 the limiting distribution is instead identified (in previous works [GW-Final]) as a random mixture of Gaussian laws—a critical multiplicative chaos measure.
• Deterministic normalization suffices in the short-interval regime: The conditional variance of $n$1 is shown to concentrate on a deterministic value, contrary to the long-sum regime where randomness persists in the normalization.

Implications and Future Directions

This work provides a complete CLT for Steinhaus random multiplicative functions in all short intervals, resolving a central problem in probabilistic number theory. The techniques introduce a blend of analytic number theory, probability, and tools from statistical mechanics (barrier events and chaos methods). Potential implications and directions include:

• Extensions to Rademacher and other random multiplicative models: The authors discuss that, while additional technical work is needed for the Rademacher case due to the lack of translation invariance, the main methods are expected to transfer.
• Connections to L-function random models: The analysis parallels results for intervals of the von Mangoldt function and squarefree numbers, suggesting universality phenomena in log-correlated random fields and random matrix theory (cf. [MS04], [GMR]).
• Multiplicative chaos and extremes: The change from Gaussian limits to heavy-tailed extreme value limits at large $n$2 is intimately tied to the physics of log-correlated fields [HarperLow, GW-Final].
• Potential for addressing deterministic function analogues: The probabilistic insight and the technical machinery—especially variance concentration under conditioning and barrier-based control—may guide analogous results for deterministic arithmetic functions (e.g., character sums, divisor functions).

Conclusion

The paper rigorously determines the limiting behavior of partial sums of Steinhaus random multiplicative functions in short intervals under the correct normalization across the full range $n$3 and $n$4. By integrating advanced martingale CLT techniques, barrier event methods from Gaussian process theory, and deep properties of random Euler products, the authors demonstrate that Gaussian fluctuation is robust in a decisive short interval regime, but breaks down as the interval length approaches the entire range.

This work clarifies the universal and phase-transitional properties of random multiplicative functions in the short interval setting, situates them within modern multiplicative chaos theory, and offers detailed analytical methods that are likely to find further applications in both probabilistic number theory and the study of log-correlated random fields.