Existence and spectrality of infinite convolutions generated by infinitely many admissible pairs

In this paper, we study the spectrality of infinite convolutions generated by infinitely many admissible pairs which may not be compactly supported, where the spectrality means the corresponding square integrable function space admits a family of exponential functions as an orthonormal basis. First, we prove that the infinite convolution exists and is a spectral measure if the sequence of admissible pairs satisfies the remainder bounded condition, and it has a subsequence consisting of general consecutive sets. Then we show that the subsequence of general consecutive sets may be replaced by a general assumption, named $\theta$-bounded condition. Finally, we investigate the infinite convolutions generated by special subsequences, and give a sufficient condition for the spectrality of such infinite convolutions.