Generating Poisson Processes by Quasi-monte Carlo

Poisson processes on xed intervals are imbedded in simulations where the goal is to estimate the expectation of some function of the process and of other random variables. Quasi-Monte Carlo is used, in an eecent way, to generate stage-wise (in tree-like fashion) certain non-consecutive arrival epochs with well-spaced indices while the others are then generated as needed by standard Monte Carlo using a numerically-stable variant of a well-known one-pass algorithm. The arrival epochs are generated conditional on the number of arrivals in the interval. That number can be generated in a stratiied way. The arrival-epoch indices are recursively selected as the respective medians of successive indices from the preceding stage, and then the corresponding arrival epochs are generated (preferably by inversion) from beta distributions corresponding to respective order statistics. Thus, the variables are assigned to coordinate indices in quasi-Monte Carlo in decreasing order of their respective importance. Alternative procedures based on recursively splitting at respective midpoints of intervals, as well as a multi-dimensional counterpart, are considered. Finally, an analog of the index-assignment property is obtained (under certain conditions) when a subproblem can be reexpressed (in a non-circular way) as generating a random permutation of order statistics, where the unordered variates are not necesarily uniforms.