论文标题
自我相似的免费添加流程和自由自制分布
Selfsimilar free additive processes and freely selfdecomposable distributions
论文作者
论文摘要
在Fan \ cite {F06}的论文中，他介绍了非共同随机过程的边际自我相似性，并证明了自由独立增量的自相似过程的边际分布是自由独立的自由学。在本文中，我们首先通过非共同随机过程的线性组合提出了一个新的定义，比范的一般一个更强的定义，尽管它们的两个定义对于具有自由独立的增量的非共同随机过程都是等效的。其次，我们证明了粉丝结果的相反，以完成自相似的自由添加过程与自由自我兼容分布之间的关系。此外，我们构建了随机积分相对于自由驱动的自由l {é} vy过程的自由添加过程的随机积分。还提供了自由累积的累积变换，并讨论了几个示例。
In the paper by Fan\cite{F06}, he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan's one in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although their two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan's result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free L{é}vy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free L{é}vy processes in terms of their free cumulant transforms is also given, and several examples are discussed.
