The best known result about the joint distribution of the backward and forward recurrence times in a renewal process concerns the asymptotic behavior for the tail of that bivariate distribution. In the present paper we study the joint behavior of the recurrence times at a fixed time point t, and we obtain both exact results and bounds for their joint tail behavior. We also obtain results about the joint moments of these two random variables and we show in particular that the expectation of the product between the two recurrence times increases with time when the interarrival distribution has a decreasing failure rate. The results are illustrated by some numerical examples.

关于续签过程中向前和向后递归时间的联合分布的最著名的结果涉及该双变量分布的尾部的渐近行为。在本文中，我们研究了在固定时间点t处重复时间的联合行为，并获得了精确的结果和联合尾部行为的界限。我们还获得了关于这两个随机变量的联合矩的结果，并且特别表明，当到达间隔分布的故障率降低时，两个递归时间之间乘积的期望随时间增加。通过一些数值示例说明了结果。