  We report some properties of heavy-tailed Sibuya-like distributions related
to thinning, self-decomposability and branching processes. Extension of the
thinning operation of on-negative integer-valued random variables to scaling by
arbitrary positive number leads to a new class of probability distributions
with generating function $Q(w)$ expressible as a Laplace transform
$\varphi(1-w)$ and probability mass function $p_n$ satisfying simple one step
recurrence relation between $p_{n+1}$ and $p_n$. We show that the compound
Poisson-Sibuya and the shifted Sibuya distributions belong to this class. Using
the fact that the same Markov property is present in stationary solutions of
the birth and death equations we identify the Sibuya distribution and some of
its variants as particular solutions of these equations. We also establish
condition when integer-valued non-negative heavy-tailed random variable has
finite $r$-th absolute moment ($0 < r < a < 1$).
