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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.15744 |
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| _version_ | 1866916702126604288 |
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| author | Miao, Junjie Liu, Hongyi Zhao, Hongbo |
| author_facet | Miao, Junjie Liu, Hongyi Zhao, Hongbo |
| contents | In this paper, we construct a class of random measures $μ^{\mathbf{n}}$ by infinite convolutions. Given infinitely many admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{\infty}$ and a positive integral sequence $\boldsymbol{n}=\{n_{k}\}_{k=1}^{\infty}$, for every $\boldsymbolω\in \mathbb{N}^{\mathbb{N}}$, we write $μ^{\mathbf{n}}(\boldsymbolω) = δ_{N_{ω_{1}}^{-n_{1}}B_{ω_{1}}} * δ_{N_{ω_{1}}^{-n_{1}}N_{ω_{2}}^{-n_{2}}B_{ω_{2}}} * \cdots$. If $n_{k}=1$ for $k\geq 1$, write $μ(\boldsymbolω)=μ^{\mathbf{n}}(\boldsymbolω)$.
First, we show that the mapping $μ^{\mathbf{n}}: (\boldsymbolω, B) \mapsto μ^{\mathbf{n}}(\boldsymbolω)(B)$ is a random measure if the family of Borel probability measures $\{μ(\boldsymbolω) : \boldsymbolω \in \mathbb{N}^{\mathbb{N}}\}$ is tight. Then, for every Bernoulli measure $\mathbb{P}$ on $\mathbb{N}^{\mathbb{N}}$, the random measure $μ^{\mathbf{n}}$ is also a spectral measure $\mathbb{P}$-a.e.. If the positive integral sequence $\boldsymbol{n}$ is unbounded, the random measure $μ^{\mathbf{n}}$ is a spectral measure regardless of the measures on the sequence space $\mathbb{N}^{\mathbb{N}}$. Moreover, we provide some sufficient conditions for the existence of the random measure $μ^{\boldsymbol{n}}$. Finally, we verify that random measures have the intermediate-value property. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and Spectrality of random measures generated by infinite convolutions Miao, Junjie Liu, Hongyi Zhao, Hongbo Functional Analysis In this paper, we construct a class of random measures $μ^{\mathbf{n}}$ by infinite convolutions. Given infinitely many admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{\infty}$ and a positive integral sequence $\boldsymbol{n}=\{n_{k}\}_{k=1}^{\infty}$, for every $\boldsymbolω\in \mathbb{N}^{\mathbb{N}}$, we write $μ^{\mathbf{n}}(\boldsymbolω) = δ_{N_{ω_{1}}^{-n_{1}}B_{ω_{1}}} * δ_{N_{ω_{1}}^{-n_{1}}N_{ω_{2}}^{-n_{2}}B_{ω_{2}}} * \cdots$. If $n_{k}=1$ for $k\geq 1$, write $μ(\boldsymbolω)=μ^{\mathbf{n}}(\boldsymbolω)$. First, we show that the mapping $μ^{\mathbf{n}}: (\boldsymbolω, B) \mapsto μ^{\mathbf{n}}(\boldsymbolω)(B)$ is a random measure if the family of Borel probability measures $\{μ(\boldsymbolω) : \boldsymbolω \in \mathbb{N}^{\mathbb{N}}\}$ is tight. Then, for every Bernoulli measure $\mathbb{P}$ on $\mathbb{N}^{\mathbb{N}}$, the random measure $μ^{\mathbf{n}}$ is also a spectral measure $\mathbb{P}$-a.e.. If the positive integral sequence $\boldsymbol{n}$ is unbounded, the random measure $μ^{\mathbf{n}}$ is a spectral measure regardless of the measures on the sequence space $\mathbb{N}^{\mathbb{N}}$. Moreover, we provide some sufficient conditions for the existence of the random measure $μ^{\boldsymbol{n}}$. Finally, we verify that random measures have the intermediate-value property. |
| title | Existence and Spectrality of random measures generated by infinite convolutions |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2504.15744 |