A non-uniform bound for translated Poisson approximation

Andrew D Barbour (Universitat Zurich)
Kwok Pui Choi (National University of Singapore)


Let $X_1, \ldots , X_n$ be independent, integer valued random variables, with $p^{\text{th}}$ moments, $p >2$, and let $W$ denote their sum. We prove bounds analogous to the classical non-uniform estimates of the error in the central limit theorem, but now, for approximation of ${\cal L}(W)$ by a translated Poisson distribution. The advantage is that the error bounds, which are often of order no worse than in the classical case, measure the accuracy in terms of total variation distance. In order to have good approximation in this sense, it is necessary for ${\cal L}(W)$ to be sufficiently smooth; this requirement is incorporated into the bounds by way of a parameter $\alpha$, which measures the average overlap between ${\cal L}(X_i)$ and ${\cal L}(X_i+1), 1 \le i \le n$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 18-36

Publication Date: February 4, 2004

DOI: 10.1214/EJP.v9-182


  1. Bikelis, A. (1966), Estimates of the remiander in the central limit theorem (in Russian), Litovsk. Mat. Sb. , 6, 323-346. MR 35 #1067
  2. Barbour, A. D. and v{C}ekanaviv cius, V. (2002), Total variation asymptotics for sums of independent integer random variables, Ann. Probab. 30, 509-545. MR 2003g: 60072
  3. Barbour, A. D., Holst, L. and Janson, S. (1992), Poisson Approximation, Oxford Studies in Probability 2, Clarendon Press, Oxford. MR 93g:60043
  4. Barbour, A. D. and Jensen, J. L. (1989), Local and tail approximations near the Poisson limit, Scand. J. Statist.16, 75-87. MR 91a:60057
  5. Barbour, A. D. and Xia, A. (1999), Poisson perturbations. ESAIM: P & S3, 131-150. MR 2000j:60026
  6. Cekanaviv cius, V. and Vaitkus, P. (2001), Centered Poisson approximation by Stein method, (in Russian) Liet. Mat. Rink.41, 409-423; translation in Lith. Math. J.41, (2001), 319-329. MR 2003e:62031
  7. Chen, L. H. Y. and Suan, I. (2003), Nonuniform bounds in discretized normal approximations, manuscript. Math. Review number not available.
  8. Chen, L. H. Y. and Shao, Q. (2001), A non-uniform Berry-Esseen bound via Stein's method, Probab. Theory Related Fields120, 236-254. MR 2002h:60037
  9. Chen, L. H. Y. and Shao, Q. (2003), Normal approximation under local dependence, manuscript. Math. Review number not available.
  10. Lindvall, T. (1992), Lectures on the coupling method, Wiley, New York. MR 94c:60002
  11. Petrov, V. V. (1975), Sums of independent random variables, Springer-Verlag, Berlin. MR 52 #9375

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.