Kybernetika 55 no. 5, 802-808, 2019

A note on discriminating Poisson processes from other point processes with stationary inter arrival times

Gusztáv Morvai and Benjamin WeissDOI: 10.14736/kyb-2019-5-0802


We give a universal discrimination procedure for determining if a sample point drawn from an ergodic and stationary simple point process on the line with finite intensity comes from a homogeneous Poisson process with an unknown parameter. Presented with the sample on the interval $[0,t]$ the discrimination procedure $g_t$, which is a function of the finite subsets of $[0,t]$, will almost surely eventually stabilize on either POISSON or NOTPOISSON with the first alternative occurring if and only if the process is indeed homogeneous Poisson. The procedure is based on a universal discrimination procedure for the independence of a discrete time series based on the observation of a sequence of outputs of this time series.


Point processes




  1. D. J. Daley and D. Vere-Jones: An introduction to the theory of point processes. Vol. II. General theory and structure. Second edition. In: Probability and its Applications. Springer, New York 2008.   DOI:10.1007/978-0-387-49835-5
  2. J. Haywood and E. Khmaladze: On distribution-free goodness-of-fit testing of exponentiality. J. Econometr. 143 (2008), 5-18.   DOI:10.1016/j.jeconom.2007.08.005
  3. O. Kallenberg: Foundations of modern probability. Second edition. In: Probability and its Applications. Springer-Verlag, New York 2002.   DOI:10.1007/978-1-4757-4015-8
  4. P. A. W. Lewis: Some results on tests for Poisson processes. Biometrika 52 (1965), 1 and 2, 67-77.   DOI:10.1093/biomet/52.1-2.67
  5. P. Massart: The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990), 3, 1269-1283.   DOI:10.1214/aop/1176990746
  6. G. Morvai and B. Weiss: Testing stationary processes for independence. Ann. Inst. H. Poincare' Probab. Statist. 47 (2011), 4, 1219-1225.   DOI:10.1214/11-aihp426
  7. B. Ryabko and J. Astola: Universal codes as a basis for time series testing. Statist. Methodol. 3 (2006), 375-397.   DOI:10.1016/j.stamet.2005.10.004
  8. H. Thorisson: Coupling, stationarity, and regeneration. In: Probability and its Applications. Springer-Verlag, New York 2000.   DOI:10.1007/978-1-4612-1236-2