On the limiting distribution of sums of random multiplicative functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912585283010560 |
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| author | Gorodetsky, Ofir Wong, Mo Dick |
| author_facet | Gorodetsky, Ofir Wong, Mo Dick |
| contents | We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}α(n)$ where $α$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved by applying the martingale central limit theorem to a suitably truncated sum. This truncation is inspired by work of Najnudel, Paquette, Simm and Vu on subcritical holomorphic multiplicative chaos setting, but analysed with a different conditioning argument generalised from Harper's work on fractional moments to circumvent integrability issues at criticality.
A significant part of the proof is devoted to the convergence in probability of the associated partial Euler product to a critical multiplicative chaos measure, independent of the mild shift away from the critical line. Our approach to the universality of critical non-Gaussian multiplicative chaos bypasses the barrier analysis with the help of a modified second moment method, and employs a novel argument based on coupling and homogenisation by change of measure, which could be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the limiting distribution of sums of random multiplicative functions Gorodetsky, Ofir Wong, Mo Dick Number Theory Complex Variables Functional Analysis Probability We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}α(n)$ where $α$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved by applying the martingale central limit theorem to a suitably truncated sum. This truncation is inspired by work of Najnudel, Paquette, Simm and Vu on subcritical holomorphic multiplicative chaos setting, but analysed with a different conditioning argument generalised from Harper's work on fractional moments to circumvent integrability issues at criticality. A significant part of the proof is devoted to the convergence in probability of the associated partial Euler product to a critical multiplicative chaos measure, independent of the mild shift away from the critical line. Our approach to the universality of critical non-Gaussian multiplicative chaos bypasses the barrier analysis with the help of a modified second moment method, and employs a novel argument based on coupling and homogenisation by change of measure, which could be of independent interest. |
| title | On the limiting distribution of sums of random multiplicative functions |
| topic | Number Theory Complex Variables Functional Analysis Probability |
| url | https://arxiv.org/abs/2508.12956 |