论文标题
广义计数过程：其非殖民和时间变化的版本
Generalized Counting Process: its Non-Homogeneous and Time-Changed Versions
论文作者
论文摘要
我们介绍了广义计数过程（GCP）的非均匀版本，即非均匀的广义计数过程（NGCP）。我们通过独立的逆稳定下属的时间来改变NGCP，以获取其分数版本，并将其称为非均匀的广义分数计数过程（NGFCP）。 NGFCP的概括是通过用独立的逆下属改变NGCP来获得的。我们得出了管理NGCP，NGFCP及其概括的边际分布的管理差分综合方程的系统。然后，我们考虑由多稳态下属改变的GCP时间，并获得其Lévy度量，相关的Bernštein函数和第一个通道时间的分布。 GCP及其分数版本，即在Lévy下属变化时的广义分数计数过程，称为时间变化的广义计数过程-I（TCGCP-I），分别为随时间变化的广义分数分数计数Process-I（TCGFCP-I）（TCGFCP-I）。我们获得了TCGCP-I的第一个通道时间和相关方程的分布。讨论了TCGCP-I对破坏理论的应用。我们从样本中获得了$ K $ TH订单统计的条件分布，该样品的大小是由TCGFCP-I的特定情况模拟的，即时间分数负二项式过程。后来，我们考虑了TCGCP-I的分数版本，并获得控制其状态概率的微分方程系统。获得其平均值，方差，协方差，{\ it等}，并使用其长期依赖性属性的使用。获得了两种特殊情况的一些结果。
We introduce a non-homogeneous version of the generalized counting process (GCP), namely, the non-homogeneous generalized counting process (NGCP). We time-change the NGCP by an independent inverse stable subordinator to obtain its fractional version, and call it as the non-homogeneous generalized fractional counting process (NGFCP). A generalization of the NGFCP is obtained by time-changing the NGCP with an independent inverse subordinator. We derive the system of governing differential-integral equations for the marginal distributions of the increments of NGCP, NGFCP and its generalization. Then, we consider the GCP time-changed by a multistable subordinator and obtain its Lévy measure, associated Bernštein function and distribution of the first passage times. The GCP and its fractional version, that is, the generalized fractional counting process when time-changed by a Lévy subordinator are known as the time-changed generalized counting process-I (TCGCP-I) and the time-changed generalized fractional counting process-I (TCGFCP-I), respectively. We obtain the distribution of first passage times and related governing equations for the TCGCP-I. An application of the TCGCP-I to ruin theory is discussed. We obtain the conditional distribution of the $k$th order statistic from a sample whose size is modelled by a particular case of TCGFCP-I, namely, the time fractional negative binomial process. Later, we consider a fractional version of the TCGCP-I and obtain the system of differential equations that governs its state probabilities. Its mean, variance, covariance, {\it etc.} are obtained and using which its long-range dependence property is established. Some results for its two particular cases are obtained.
