Quasi-infinitely divisible distributions

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A quasi-infinitely divisible distribution is a probability distribution whose characteristic function admits a Lévy-Khintchine type representation, however with a signed Lévy measure (the quasi-Lévy measure) rather than with a Lévy measure. Equivalently, a probability distribution is quasi-infinitely divisible if its characteristic function is the quotient of the characteristic functions of two infinitely divisible distributions. Quasi-infinitely divisible distributions appear naturally in the factorisation problem of infinitely divisible distributions. While infinitely divisible distributions form a well-studied class of probability distributions, much less is known about quasi-infinitely divisible distributions, and a systematic study of these distributions has only been initiated recently.The aim of this project is to deepen the understanding of quasi-infinitely divisible distributions. In particular, we intend to find conditions ensuring quasi-infinite divisibility of given distributions, and for a given quasi-infinitely distribution, to study its properties in terms of the quasi-Lévy measure. While much of the existing literature on quasi-infinitely divisible distributions at the moment is concerned only with the univariate case, we intend to study multivariate quasi-infinitely divisible distributions and in particular study if a Cramér-Wold device holds for this class of distributions. We shall also look for a natural connection of quasi-infinitely divisible distributions to stochastic processes.