Martingale central limit theorem for random multiplicative functions
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917686069428224 |
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| author | Gorodetsky, Ofir Wong, Mo Dick |
| author_facet | Gorodetsky, Ofir Wong, Mo Dick |
| contents | Let $α$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}α(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian limiting distribution. More precisely, we establish a generalised central limit theorem with random variance determined by the total mass of a random measure associated with $αf$.
Our result applies to $d_z$, the $z$-th divisor function, as long as $z$ is strictly between $0$ and $\tfrac{1}{\sqrt{2}}$. Other examples of admissible $f$-s include any multiplicative indicator function with the property that $f(p)=1$ holds for a set of primes of density strictly between $0$ and $\tfrac{1}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Martingale central limit theorem for random multiplicative functions Gorodetsky, Ofir Wong, Mo Dick Number Theory Probability Let $α$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}α(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian limiting distribution. More precisely, we establish a generalised central limit theorem with random variance determined by the total mass of a random measure associated with $αf$. Our result applies to $d_z$, the $z$-th divisor function, as long as $z$ is strictly between $0$ and $\tfrac{1}{\sqrt{2}}$. Other examples of admissible $f$-s include any multiplicative indicator function with the property that $f(p)=1$ holds for a set of primes of density strictly between $0$ and $\tfrac{1}{2}$. |
| title | Martingale central limit theorem for random multiplicative functions |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2405.20311 |