On the quality of poisson approximations

  • Ward Whitt
Article

Summary

Poisson processes (possibly nonhomogeneous) are constructed in the function spaces D q ≡D([0, 1] q , R) and Dq q x ⋯ x D q in order to approximate superpositions of uniformly sparse point processes and partial sums of infinitesimal integer-valued nonnegative random variables. Bounds for the Prohorov distance are computed, where the Prohorov distance is defined on the space of all probability measures on D q, with the Skorohod metric being used on D q. These bounds yield functional central limit theorems (invariance principles) and rates of convergence for functional central limit theorems involving convergence to the Poisson process. In this regard, this paper is an extension of Section 6 of Dudley [4].

For related references and general background, the reader is referred to: Billingsley [2] for convergence of probability measures; \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{C} \)inlar [3] for superpositions of point processes; Dudley [4] for rates of convergence; Bickel and Wichura [1], Neuhaus [6], and Straf [7] for the space D q; and Whitt [8] for methods to apply the rates of convergence results here to obtain rates of convergence for related functionals and processes.

This paper is organized as follows. Partial sums of integer-valued random variables are treated in Section 1; superpositions of point processes are treated in Section 2; and the extension of the results in Section 2 to the superposition of p-dimensional (1≦p≦∞) point processes is outlined in Sections. An intuitive understanding of the results can perhaps be achieved more quickly by examining the special cases in Examples 1.1 and 2.1.

Keywords

Poisson Process Point Process Random Element Invariance Principle Discrete Approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bickel, P.J., Wichura, M.J.: Convergence criteria for multi-parameter stochastic processes and some applications. Ann. Math. Statist. 42, 1656–1670 (1971)MATHMathSciNetGoogle Scholar
  2. 2.
    Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968MATHGoogle Scholar
  3. 3.
    Cinlar, E.: Superposition of point processes. Stochastic Point Processes: Statistical Analysis, Theory and Applications, Ed. P.A.W. Lewis, New York: John Wiley and Sons 549–606, 1972Google Scholar
  4. 4.
    Dudley, R. M.: Speeds of metric probability convergence. Z. Wahrscheinlichkeitstheorie verw. Gebiete 22, 323–332 (1972)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables, Second Edition. Reading, Massachusetts: Addison-Wesley 1968MATHGoogle Scholar
  6. 6.
    Neuhaus, G.: On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist. 42, 1285–1295 (1971)MATHMathSciNetGoogle Scholar
  7. 7.
    Straf, M.L.: Weak convergence of stochastic processes with several parameters. Proc. Sixth Berk. Symp. Stat. Prob. 2, 187–221 (1972)MATHMathSciNetGoogle Scholar
  8. 8.
    Whitt, W.: Preservation of rates of convergence under mappings. Z. Wahrscheinlichkeitstheorie verw. Gebiete. To appearGoogle Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • Ward Whitt
    • 1
  1. 1.Department of Administrative SciencesYale UniversityNew HavenUSA

Personalised recommendations