Existence and Spectrality of random measures generated by infinite convolutions

Abstract

In this paper, we construct a class of random measures $\hat{I}{¼}^{\mathbf{n}}$ by infinite convolutions. Given infinitely many admissible pairs $\{(N_k,B_k){\}}_{k=1}^{\infty }$ and a positive integral sequence $\mathbf{n}=\{n_k{\}}_{k=1}^{\infty }$, for every $\ddot{\mathbf{I}}‰\in {\mathbb{N}}^{\mathbb{N}}$, we write $\hat{I}{¼}^{\mathbf{n}}(\ddot{\mathbf{I}}‰)=\hat{I}{´}_{N_{\ddot{I}{‰}_1}^{-n_1}{B}_{\ddot{I}{‰}_1}}\ast \hat{I}{´}_{N_{\ddot{I}{‰}_1}^{-n_1}{N}_{\ddot{I}{‰}_2}^{-n_2}{B}_{\ddot{I}{‰}_2}}\ast \cdots$. If $n_k=1$ for $k\ge 1$, write $\hat{I}¼(\ddot{\mathbf{I}}‰)=\hat{I}{¼}^{\mathbf{n}}(\ddot{\mathbf{I}}‰)$. First, we show that the mapping $\hat{I}{¼}^{\mathbf{n}}:(\ddot{\mathbf{I}}‰,B)\mapsto \hat{I}{¼}^{\mathbf{n}}(\ddot{\mathbf{I}}‰)(B)$ is a random measure if the family of Borel probability measures $\{\hat{I}¼(\ddot{\mathbf{I}}‰):\ddot{\mathbf{I}}‰\in {\mathbb{N}}^{\mathbb{N}}\}$ is tight. Then, for every Bernoulli measure $\mathbb{P}$ on ${\mathbb{N}}^{\mathbb{N}}$, the random measure $\hat{I}{¼}^{\mathbf{n}}$ is also a spectral measure $\mathbb{P}$-a.e.. If the positive integral sequence $\mathbf{n}$ is unbounded, the random measure $\hat{I}{¼}^{\mathbf{n}}$ 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 $\hat{I}{¼}^{\mathbf{n}}$. Finally, we verify that random measures have the intermediate-value property.