We develop a general framework for stationary marked point processes in discrete time. We start with a careful analysis of the sample paths. Our initial representation is a sequence $$\{(t_j,k_j): j\in {\mathbb {Z}}\}$${(tj,kj):j∈Z} of times $$t_j\in {\mathbb {Z}}$$tj∈Z and marks $$k_j\in {\mathbb {K}}$$kj∈K, with batch arrivals (i.e., $$t_j=t_{j+1}$$tj=tj+1) allowed. We also define alternative interarrival time and sequence representations and show that the three different representations are topologically equivalent. Then, we develop discrete analogs of the familiar stationary stochastic constructs in continuous time: time-stationary and point-stationary random marked point processes, Palm distributions, inversion formulas and Campbell’s theorem with an application to the derivation of a periodic-stationary Little’s law. Along the way, we provide examples to illustrate interesting features of the discrete-time theory.

中文翻译：

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离散时间中的标记点过程

我们为离散时间的静止标记点过程开发了一个通用框架。我们首先仔细分析样本路径。我们的初始表示是一个序列 $$\{(t_j,k_j): j\in {\mathbb {Z}}\}$${(tj,kj):j∈Z} 次 $$t_j\in {\ mathbb {Z}}$$tj∈Z 并标记 $$k_j\in {\mathbb {K}}$$kj∈K，批到达（即 $$t_j=t_{j+1}$$tj= tj+1) 允许。我们还定义了替代的到达间隔时间和序列表示，并表明这三种不同的表示在拓扑上是等效的。然后，我们开发了连续时间中熟悉的平稳随机构造的离散模拟：时间平稳和点平稳随机标记点过程、Palm 分布、反演公式和坎贝尔定理，并应用于推导周期平稳的利特尔定律。一路上，