  We consider discrete probability laws on the real line, whose characteristic
functions are separated from zero. In particular, this class includes arbitrary
discrete infinitely divisible laws and lattice probability laws, whose
characteristic functions have no zeroes on the real line. We show that
characteristic functions of such laws admit spectral L\'evy--Khinchine type
representations. We also apply the representations of such laws to obtain limit
and compactness theorems with convergence in variation to probability laws from
this class. Thus we generalize the corresponding results from the recent papers
by Lindner, Pan, Sato, and by Khartov.
