多维迪克曼分布与算子自分解性
Multidimensional Dickman distribution and operator selfdecomposability
AI总结：
本文研究多维迪克曼分布的定义及其在多维莱维过程小跳跃近似中的应用，证明其无限可分性和算子自分解性，并识别其作为极限分布的几种情况。
摘要：
一维迪克曼分布在数论、组合学、物理和生物学中的随机模型中普遍存在。最近文献中出现了一种多维迪克曼分布的定义，并应用于近似多维莱维过程的小跳跃。本文将此定义扩展到向量值随机元的类，将其特征化为特定仿射变换的固定点，该变换涉及由均匀分布随机变量的矩阵指数得到的随机矩阵。我们证明这些新分布具有无限可分性和算子自分解性。此外，我们识别了几种情况下该新分布作为极限分布出现的情况。
英文摘要：
The one-dimensional Dickman distribution arises in various stochastic models across number theory, combinatorics, physics, and biology. Recently, a definition of the multidimensional Dickman distribution has appeared in the literature, together with its application to approximating the small jumps of multidimensional Lévy processes. In this paper, we extend this definition to a class of vector-valued random elements, which we characterise as fixed points of a specific affine transformation involving a random matrix obtained from the matrix exponential of a uniformly distributed random variable. We prove that these new distributions possess the key properties of infinite divisibility and operator selfdecomposability. Furthermore, we identify several cases where this new distribution arises as a limiting distribution.
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