We consider a new class
of distribution functions
that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution
functions
and
such that
. A distribution
function of class
is quasi-infinitely divisible in the sense that its characteristic function admits the
Lévy-type representation with a “signed spectral measure”. This class is a wide natural
extension of the fundamental class of infinitely divisible distribution functions and it is
being actively studied now. We are interested in conditions for a distribution function
to belong to class
for the unexplored
case, where
may
have a continuous singular part. We propose a criterion under the assumption that the continuous
singular part of
is dominated by the discrete part in a certain sense. The criterion generalizes the
previous results by Alexeev and Khartov for discrete probability laws and the results
by Berger and Kutlu for the mixtures of discrete and absolutely continuous laws. In
addition, we describe the characteristic triplet of the corresponding Lévy-type
representation, which may contain a continuous singular part. We also show that the
assumption of the dominated continuous singular part cannot be omitted or even
slightly extended (without some special assumptions). We apply the general
criterion to some interesting particular examples. We also positively solve the
decomposition problem stated by Lindner, Pan and Sato within the case being
considered.
