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Optimal Poisson approximation of uniform empirical processes

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  • Adell, JoséA.
  • de la Cal, Jesús

Abstract

In this paper, we discuss the optimality of Poisson approximation of uniform empirical processes of size n in a small interval [0, l], in the sense that the sup-norm distance between their paths has minimum expectation. Two optimal constructions are considered. The first one depends on [0, l] and makes sense if and only if l = o(n-1/2), whereas the second one does not, and makes sense if and only if l = o(n-1). In both cases, we obtain the exact probability that the paths of the two processes coincide on [0, l] as well as, under appropriate assumptions, the exact order of convergence of the tail probabilities concerning the sup-norm distance between their paths. We use elementary coupling techniques which allow us to give short and simple proofs.

Suggested Citation

  • Adell, JoséA. & de la Cal, Jesús, 1996. "Optimal Poisson approximation of uniform empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 135-142, November.
    • Handle: RePEc:eee:spapps:v:64:y:1996:i:1:p:135-142
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    Cited by:

    1. Ruzankin, Pavel S. & Borisov, Igor S., 2020. "On the rate of Poisson approximation to Bernoulli partial sum processes," Statistics & Probability Letters, Elsevier, vol. 162(C).

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