A framework for analyzing the periodically-observed time-homogeneous Poisson process
针对实际系统中事件到达时间被周期记录（如小时、天）导致的离散化问题，提出了周期观测时间齐次泊松过程模型，并推导了其极限分布和矩，通过模拟实验发现某些离散化水平下传统检验无法识别该过程。
When counting process data are collected from real-world systems, the arrival of each event is often reported in periodic time units (e.g., hour, day, week, month) and the precise arrival time is lost. This periodic reporting introduces discretization error into arrival time data, fundamentally changing the resulting interarrival distribution and inhibiting comparisons to continuous-time stochastic processes (e.g., Poisson process). This article formulates the periodically-observed time-homogeneous Poisson process (PTPP) to account for discretization due to periodic observation when the underlying system is a time-homogeneous Poisson process. In contrast with the analogous Poisson process, the PTPP is not a renewal process; however, its arrivals can be modeled by an infinite-state discrete-time Markov chain with two state variables: the recorded interarrival time and order of the event within the current observation period. The marginal limiting distribution for the first variable (i.e., the limiting interarrival distribution) is derived along with its cumulative distribution, moment generating function, first two moments, and variance. This article shows, through a simulation-based experiment, that there exist a range of discretization-levels for which neither the interarrival nor counting distribution can effectively identify a periodically-observed Poisson process through goodness-of-fit testing; the PTPP model bridges this gap.