Time intervals and counting in point processes

Time point processes can be analyzed in two different ways: by the number of points in arbitrary time intervals or by distance between points. This corresponds to two distinct physical devices: counting or time-interval measurements. We present an explicit calculation, valid for arbitrary regular processes, of the statistical properties of time intervals such as residual or life time in terms of counting probabilities. For this calculation, we show that these intervals must be considered as random variables defined by conditional distributions.