Existence and Spectrality of random measures generated by infinite convolutions
Junjie Miao, Hongyi Liu, Hongbo Zhao
TL;DR
The paper develops a framework for random measures generated by infinite convolutions and analyzes their spectrality. By structuring the problem around admissible pairs and Hadamard-triple conditions, it shows that under tightness of the induced measure family, the random-convolution construction yields well-defined random measures μ^{ }, which are spectral in various regimes: unbounded n gives spectral realizations for all ω, and Bernoulli-base randomness yields spectrality almost surely. It provides practical sufficient conditions (e.g., RBC and growth bounds) guaranteeing existence and tightness, and demonstrates a robust intermediate-value property: the dimension dim_H μ can be tuned across a range by adjusting the underlying Bernoulli weights, with explicit formulas. The results reveal rich spectral and geometric structures in fractal-like random measures and extend the spectral theory of infinite convolutions to a probabilistic setting with concrete, verifiable criteria.
Abstract
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.