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A Compound Poisson Convergence Theorem for Sums of \(m\)-Dependent Variables

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Abstract

We prove the Simons–Johnson theorem for sums \(S_n\) of \(m\)-dependent random variables with exponential weights and limiting compound Poisson distribution \(\mathrm {CP}(s,\lambda )\). More precisely, we give sufficient conditions for \(\sum _{k=0}^\infty {\mathrm e}^{hk}\vert P(S_n=k)-\mathrm {CP}(s,\lambda )\{k\}\vert \rightarrow 0\) and provide an estimate on the rate of convergence. It is shown that the Simons–Johnson theorem holds for the weighted Wasserstein norm as well. The results are then illustrated for \(N(n;k_1,k_2)\) and \(k\)-runs statistics.

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References

  1. Barbour, A.D., Chen, L.H.Y., Choi, K.P.: Poisson approximation for unbounded functions I: independent summands. Stat. Sinica 5, 749–766 (1995)

    MATH  MathSciNet  Google Scholar 

  2. Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Clarendon Press, Oxford (1992)

    MATH  Google Scholar 

  3. Bergström, H.: On asymptotic expansion of probability functions. Skand. Aktuar. 1, 1–34 (1951)

    Google Scholar 

  4. Borisov, I.S., Ruzankin, P.S.: Poisson approximation for expectations of unbounded functions of independent random variables. Ann. Probab. 30, 1657–1680 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Čekanavičius, V.: On the convergence of Markov binomial to Poisson distribution. Stat. Probab. Lett. 58, 83–91 (2002)

    Article  MATH  Google Scholar 

  6. Chen, L.H.Y.: On the convergence of Poisson binomial to Poisson distributions. Ann. Probab. 2, 178–180 (1974)

    Article  MATH  Google Scholar 

  7. Chen, L.H.Y.: An approximation theorem for convolutions of probability measures. Ann. Probab. 3, 992–999 (1975)

    Article  MATH  Google Scholar 

  8. Chen, L.H.Y., Roos, M.: Compound Poisson approximation for unbounded functions on a group with application to large deviations. Probab. Theory Relat. Fields 103, 515–528 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dasgupta, R.: Nonuniform rates of convergence to the Poisson distribution. Sankhya. Ser. A. 54, 460–463 (1992)

    MATH  MathSciNet  Google Scholar 

  10. Dafnis, S.D., Antzoulakos, D.L., Philippou, A.N.: Distributions related to \((k_1; k_2)\) events. J. Stat. Plan. Inference 140, 16911700 (2010)

    Article  MathSciNet  Google Scholar 

  11. Heinrich, L.: A method for the derivation of limit theorems for sums of m-dependent random variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete 60, 501–515 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Huang, W.T., Tsai, C.S.: On a modified binomial distribution of order k. Stat. Probab. Lett. 11, 125–131 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. O’Brien, P.C., Dyck, P.J.: A runs test based on run lengths. Biometrics 41, 237–244 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  14. Presman, E.L.: Approximation in variation of the distribution of a sum of independent Bernoulli variables with a Poisson law. Theory Probab. Appl. 30(2), 417–422 (1986)

    Article  MATH  Google Scholar 

  15. Röllin, A.: Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli 11, 1115–1128 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ruzankin, P.S.: Approximation for expectations of unbounded functions of dependent integer-valued random variables. J. Appl. Probab. 47, 594–600 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shiryaev, A.N.: Probability (Graduate Texts in Mathematics), vol. 95, 2nd edn. Springer, Berlin (1995)

    Google Scholar 

  18. Simons, G., Johnson, N.L.: On the convergence of binomial to Poisson distributions. Ann. Math. Stat. 42, 1735–1736 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  19. Vellaisamy, P.: Poisson approximation for \((k_1, k_2)\) events via the Stein-Chen method. J. Appl. Probab. 41, 1081–1092 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wang, Y.H.: A compound Poisson convergence theorem. Ann. Probab. 19, 452–455 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang, X., Xia, A.: On negative binomial approximation to k-runs. J. Appl. Probab. 45, 456–471 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors wish to thank the referees for helpful comments which helped to improve the paper.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 , Vilnius, Lithuania

    V. Čekanavičius

  2. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    P. Vellaisamy

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  1. V. Čekanavičius
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  2. P. Vellaisamy
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Correspondence to V. Čekanavičius.

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Čekanavičius, V., Vellaisamy, P. A Compound Poisson Convergence Theorem for Sums of \(m\)-Dependent Variables. J Theor Probab 28, 1145–1164 (2015). https://doi.org/10.1007/s10959-014-0540-5

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  • DOI: https://doi.org/10.1007/s10959-014-0540-5

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