# 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

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### References

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