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We report some properties of heavy-tailed Sibuya-like distributions relatedto thinning, self-decomposability and branching processes. Extension of thethinning operation of on-negative integer-valued random variables to scaling byarbitrary positive number leads to a new class of probability distributionswith generating function $Q(w)$ expressible as a Laplace transform$\varphi(1-w)$ and probability mass function $p_n$ satisfying simple one steprecurrence relation between $p_{n+1}$ and $p_n$. We show that the compoundPoisson-Sibuya and the shifted Sibuya distributions belong to this class. Usingthe fact that the same Markov property is present in stationary solutions ofthe birth and death equations we identify the Sibuya distribution and some ofits variants as particular solutions of these equations. We also establishcondition when integer-valued non-negative heavy-tailed random variable hasfinite $r$-th absolute moment ($0 < r < a < 1$).

On Sibuya-like distributions in branching and birth-and-death processes