A note on track-length sampling with non-exponential distributions

Overview & Synopsis

Track-length sampling is the process of sampling random intervals according to a distance distribution. It means that, instead of sampling a punctual distance from the distance distribution, track-length sampling generates an interval of possible distances. The track-length sampling process is correct if the expectation of the intervals is the target distance distribution. In other words, averaging all the sampled intervals should converge towards the distance distribution as their number increases. In this note, we emphasize that the distance distribution that is used for sampling punctual distances and the track-length distribution that is used for sampling intervals are not the same in general. This difference can be surprising because, to our knowledge, track-length sampling has been mostly studied in the context of transport theory where the distance distribution is typically exponential: in this special case, the distance distribution and the track-length distribution happens to be both the same exponential distribution. However, they are not the same in general when the distance distribution is non-exponential. We show that track-length sampling can be used with non-exponential distance distributions if they are monotonically decreasing and we derive the general expression of the track-length distribution.