Uitgedund en gerekt Poissonproces

W. Vervaat

Statistica Neerlandica, 1967, vol. 21, issue 3‐4, 245-268

Abstract: This paper generalizes an interrupted Poisson process first studied by ZAAT [8] and later by RUNNENBURG and VERVAAT [7]. The process is obtained by combining two independent stochastic processes on the non‐negative real axis. The first of these processes is a stationary Poisson process with intensity Λ and the second an alternating renewal process [1] dividing the axis in a‐ and b‐intervals alternatively. By omitting those points of the stationary Poisson process that fall in b‐intervals we create a new point process that we prefer to call a “thinned Poisson process”. The alternating renewal process is determined by the distribution functions A and B of the lengths of the a‐ and b‐intervals which are not the first interval to the right ofO and by an “initial distribution” (p, A0, B0), where p is the probability of starting with an a‐interval in 0, A0 the distribution function of the length of the first a‐interval provided that we begin with an a‐interval in 0, and B0 the distribution function of the first b‐interval provided that we begin with a b‐interval in 0. It is here shown that there always exist initial distributions for which the distribution function of the length of an interval between two successive points in the thinned Poisson process is the same for all intervals, and the class of initial distributions with this property is obtained. The common distribution function G of these lenghts has Laplace‐Stieltjes transform G as given in (3.15); it turns out that G depends on A, B and Λ only. For any initial distribution the distribution function of the distance between the nth and the (n + l)st point to the right ofO in the thinned Poisson process tends to the distribution function G as n tends to infinity. Finally the covariance of two successive distances between three successive points in the thinned Poisson process is studied. Some examples are given in which both positive and negative correlations occur.

Date: 1967
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