Existence, equivalence and spectrality of infinite convolutions in $\R^d$

TL;DR

This paper investigates the conditions under which infinite convolutions in multi-dimensional space exist, are equivalent, and possess spectral properties, using probability theory techniques to analyze their convergence and spectrality.

Contribution

It introduces the concept of equivalent sequences for infinite convolutions and proves their convergence and spectral properties are preserved under equivalence.

Findings

Infinite convolutions generated by equivalent sequences converge simultaneously.

Spectrality is preserved for infinite convolutions generated by equivalent sequences.

Provides sufficient conditions for existence and spectral properties in higher dimensions.

Abstract

In this paper, we study existence, equivalence and spectrality of infinite convolutions which may not be compactly supported in $d$-dimensional Euclidean space by manipulating various techniques in probability theory. First, we define the equivalent sequences, and we prove that the infinite convolutions converges simultaneously if they are generated by equivalent sequences. Moreover, the equi-positivity keeps unchanged for infinite convolutions generated by equivalent sequences. Next, we study the spectrality of infinite convolutions generated by admissible pairs, and we show such infinite convolutions have the same spectrum if they are generated by the equivalent sequences. Finally, we provide some sufficient conditions for the existence and spectral properties of infinite convolutions in higher dimensions.