Past intensity of a terminated Poisson process (Q789813)

This paper considers a Poisson process terminated when the number of events within a ''window'' of length w exceeds a threshold number k. The question asked is: given a collection of processes terminated in this way, what can be said of the rate, r(t), of the process at time t units preceding the termination. A solution is presented, using renewal arguments, for the case \(k=2\), which is surprisingly complicated. The results show that, although r(t) decreases for \(t<w\) (reflecting the effect of the terminating condition), r(t) is not decreasing for all t. \(r(t)>1/w\) can occur even for \(t>w\). The limiting value of r(t), as \(t\to \infty\) is also obtained, and is shown to approach a non-zero value. For \(k>2\), a renewal approach is not possible, and a solution is not available. A discrete-time approximation, however, for a general stochastic point process is referred to in one of the references. The paper clears up a fault in \textit{M. D. Maltz} and the first author, Artificial inflation of a delinquency rate by a selection artifact. Oper. Res. 28, 547-559 (1980), and complements a similar analysis recently published by \textit{L. Tierney}, ibid. 31, 852-865 (1983; Zbl 0523.92026).