  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.
