Estimating the distributions of scan statistics with high precision (Q1848514)

Let \(N(t)\) be a Poisson process with intensity \(\lambda\), let \(a,T>0\), and let \(L=T/a\) be integer. The author is interested in approximations of the probability \[ Q_L(n,\lambda)=Q_L=\Pr \Bigl\{\max_{0\leq t\leq T-a}(N(t+a)-N(t))<n \Bigr\} \] and analogous probabilities for min. The idea is to approximate \(Q_L\) for large \(L\) by \(Q_l\) with small \(l\) which can be easily computed by the Huntington-Naus exact formula [see \textit{R. J. Huntington} and \textit{J. I. Naus}, Ann. Probab. 3, 894-896 (1975; Zbl 0316.60014)]. New approximations are derived, e.g. \(Q_L\cong (2Q_2-Q_3)(1+Q_2-Q_3+2(Q_2-Q_3)^2)^{-(L-1)}\), and their relative errors are estimated. The obtained approximations are used to estimate the distribution and expectation of waiting time till a cluster. Probabilities connected with clusters on a circle and on a plain are also considered.