随机过程族的自相似性和Lamperti收敛性
Self-Similarity and Lamperti Convergence for Families of Stochastic Processes
摘要：
我们定义了一种新类型的自相似性的单参数随机过程的家庭，这适用于一些重要的家庭的过程，不是自相似的传统意义上。这包括一类新的分数Hougaard运动，定义为Hougaard Lévy过程的移动平均，以及一些著名的Hougaard Lévy过程族，如Poisson过程，带漂移的布朗运动和逆高斯过程。这类族与普通的自相似过程有许多共同的性质，包括它们的协方差函数的形式，以及它们在随机过程族的兰珀蒂型极限定理中表现为极限的事实。
英文摘要：
We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to a number of important families of processes that are not self-similar in the conventional sense. This includes a new class of fractional Hougaard motions defined as moving averages of Hougaard Lévy process, as well as some well-known families of Hougaard Lévy processes such as the Poisson processes, Brownian motions with drift, and the inverse Gaussian processes. Such families have many properties in common with ordinary self-similar processes, including the form of their covariance functions, and the fact that they appear as limits in a Lamperti-type limit theorem for families of stochastic processes.
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