<p>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 introduce remainder bounded condition (RBC), and we show it is a sufficient condition for the existence of the infinite convolution. Then we prove that the infinite convolution generated by a sequence of admissible pairs is a spectral measure if the sequence of admissible pairs satisfies RBC, and it has a subsequence consisting of general consecutive sets. Next, we show that the subsequence of general consecutive sets may be replaced by a general assumption, named partial concentration condition (PCC). Finally, we investigate the infinite convolutions generated by special subsequences, and we give sufficient conditions for the spectrality of such infinite convolutions.</p>