We introduce a new heavy tailed distribution on \(\mathbb {Z}_+\) that arises as the infinite convolution of scaled Sibuya distributions. We provide closed form expressions for its probability mass function, its cumulative distribution function, and its probability generating function. We interpret our main results in terms of the weak convergence of partial sums of a binomially thinned sequence of i.i.d. random variables with a common scaled Sibuya distribution. Properties of infinite divisibility and discrete self-decomposability of the new distribution are also discussed. As an application, we briefly describe an integer-valued autoregressive process of order one with a scaled Sibuya innovation sequence. Finally, we discuss some partial extensions of our results to the case of the generalized Sibuya distribution introduced by Kozubowski and Podgórski., Ann. of the Inst. Statist. Math., 70(4), 855-887., 2018.
