  We study spectral measures generated by infinite convolution products of
discrete measures generated by Hadamard triples, and we present sufficient
conditions for the measures to be spectral, generalizing a criterion by
Strichartz. We then study the spectral measures generated by random
convolutions of finite atomic measures and rescaling, where the digits are
chosen from a finite collection of digit sets. We show that in dimension one,
or in higher dimensions under certain conditions, "almost all" such measures
generate spectral measures, or, in the case of complete digit sets,
translational tiles. Our proofs are based on the study of self-affine spectral
measures and tiles generated by Hadamard triples in quasi-product form.
