  A quasi-infinitely divisible distribution on $\mathbb{R}$ is a probability
distribution whose characteristic function allows a L\'evy-Khintchine type
representation with a "signed L\'evy measure", rather than a L\'evy measure.
Quasi-infinitely divisible distributions appear naturally in the factorization
of infinitely divisible distributions. Namely, a distribution $\mu$ is
quasi-infinitely divisible if and only if there are two infinitely divisible
distributions $\mu_1$ and $\mu_2$ such that $\mu_1 \ast \mu = \mu_2$. The
present paper studies certain properties of quasi-infinitely divisible
distributions in terms of their characteristic triplet, such as properties of
supports, finiteness of moments, continuity properties and weak convergence,
with various examples constructed. In particular, it is shown that the set of
quasi-infinitely divisible distributions is dense in the set of all probability
distributions with respect to weak convergence. Further, it is proved that a
distribution concentrated on the integers is quasi-infinitely divisible if and
only if its characteristic function does not have zeroes, with the use of the
Wiener-L\'evy theorem on absolutely convergent Fourier series. A number of fine
properties of such distributions are proved based on this fact. A similar
characterisation is not true for non-lattice probability distributions on the
line.

Probability