Compactness criteria for quasi-infinitely divisible distributions on the integers

We consider the class of quasi-infinitely divisible distributions. These distributions have appeared before in the theory of decompositions of probability laws, and nowadays they have various applications in theory of stochastic processes, physics, and insurance mathematics. The characteristic functions of quasi-infinitely divisible distributions admit Lévy type representation with real drift, nonnegative Gaussian variance, and “signed Lévy measure”. Lindner et al. (2018) have recently done the first detailed analysis of these distributions based on such representations. The most complete results were established by them for the quasi-infinitely divisible distributions on the integers. In particular, the authors have obtained a criterion of weak convergence for distributions from this class in terms of parameters of their Lévy type representations. In the present short paper we complement this result by similar criteria of relative and stochastic compactness for quasi-infinitely divisible distributions with partial weak limits from this class. We also show that if a general sequence of distributions on the integers is relatively compact with quasi-infinitely divisible partial weak limits, then all distributions of the sequence are quasi-infinitely divisible except a finite number.