Quasi-infinitely divisible distributions

拟无穷可分分布是一种概率分布，其特征函数允许Lévy-Khintchine型表示，但具有符号Lévy测度(拟Lévy测度)而不具有Lévy测度。等价地，如果一个概率分布的特征函数是两个无穷可分分布的特征函数的商，则该概率分布是拟无穷可分的。拟无穷可分分布是在无穷可分分布的因式分解问题中自然出现的。虽然无穷可分分布形成了一类研究得很好的概率分布，但对拟无穷可分分布的研究却少之又少，而且直到最近才开始对这些分布进行系统的研究。特别地，我们打算找到保证给定分布的拟无限整除的条件，并且对于给定的拟无限分布，研究其关于拟Lévy测度的性质。虽然目前关于拟无穷可分分布的大部分文献只涉及单变量情形，但我们打算研究多元拟无穷可分分布，特别是研究Cramér-Wold装置对这类分布是否成立。我们还将寻找准无穷可分分布与随机过程之间的自然联系。

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.