Existence and Spectrality of random measures generated by infinite convolutions

In this paper, we construct a class of random measures $\mu^{\mathbf{n}}$ 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 $\boldsymbol{\omega}\in \mathbb{N}^{\mathbb{N}}$, we write $\mu^{\mathbf{n}}(\boldsymbol{\omega}) = \delta_{N_{\omega_{1}}^{-n_{1}}B_{\omega_{1}}} * \delta_{N_{\omega_{1}}^{-n_{1}}N_{\omega_{2}}^{-n_{2}}B_{\omega_{2}}} * \cdots$. If $n_{k}=1$ for $k\geq 1$, write $\mu(\boldsymbol{\omega})=\mu^{\mathbf{n}}(\boldsymbol{\omega})$. First, we show that the mapping $\mu^{\mathbf{n}}: (\boldsymbol{\omega}, B) \mapsto \mu^{\mathbf{n}}(\boldsymbol{\omega})(B)$ is a random measure if the family of Borel probability measures $\{\mu(\boldsymbol{\omega}) : \boldsymbol{\omega} \in \mathbb{N}^{\mathbb{N}}\}$ is tight. Then, for every Bernoulli measure $\mathbb{P}$ on $\mathbb{N}^{\mathbb{N}}$, the random measure $\mu^{\mathbf{n}}$ is also a spectral measure $\mathbb{P}$-a.e.. If the positive integral sequence $\boldsymbol{n}$ is unbounded, the random measure $\mu^{\mathbf{n}}$ 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 $\mu^{\boldsymbol{n}}$. Finally, we verify that random measures have the intermediate-value property.