Abstract In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.

中文翻译：

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更新过程中向后和向前递归时间的协方差：平稳情况和普通情况的渐近线

摘要 在泊松过程中，众所周知，给定时间点 t 的前向和后向循环次数是独立的随机变量。在更新过程中，虽然这些量的联合分布是已知的（渐近），但关于它们的协方差函数的结果似乎很少。在本文中，我们研究了这种协方差，特别是，我们说明了在可靠性分类和潜在相互更新和相关平衡的变异系数方面它为正、零或负的必要和充分条件分配。我们的结果既适用于稳定状态下的普通更新过程，也适用于平稳过程。