COMPOUND POISSON APPROXIMATIONS FOR THE NUMBERS OF EXTREME SPACINGS

The accuracy of the Poisson approximation to the distribution of the numbers of large and small m-spacings, when n points are placed at random on the circle, was analysed using the Stein-Chen method in Barbour et al. (1992b). The Poisson approximation for m greater-than-or-equal-to 2 was found not to be as good as for 1-spacings. In this paper, rates of approximation of these distributions to suitable compound Poisson distributions are worked out, using the CP-Stein-Chen method and an appropriate coupling argument. The rates are better than for Poisson approximation for m greater-than-or-equal-to 2, and are of order O((log n)2/n) for large m-spacings and of order O(1/n) for small m-spacings, for any fixed m greater-than-or-equal-to 2, if the expected number of spacings is held constant as n --> infinity.