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Compound Poisson approximations for sums of 1-dependent random variables. I

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Abstract

We approximate two-runs statistic by various compound Poisson distributions and second-order asymptotic expansions. The accuracy of approximation is estimated in the total-variation and local metrics. For a special case, asymptotically sharp constants are calculated. Heinrich’s method is used in the proofs.

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References

  1. R. Arratia, L. Goldstein, and L. Gordon, Poisson approximation and the Chen–Stein method, Stat. Sci., 5(4):403–434, 1990.

    MathSciNet  MATH  Google Scholar 

  2. A.D. Barbour and O. Chryssaphino, Compound Poisson approximation: A user’s guide, Ann. Appl. Probab., 11(3):964–1002, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  3. A.D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford Univ. Press, 1992.

  4. A.D. Barbour and V. Čekanavičius, Total variation asymptotics for sums of independent integer random variables, Ann. Probab., 30(2):509–545, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  5. A.D. Barbour and A. Xia, Poisson perturbations, ESAIM, Probab. Stat., 3:131–150, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  6. T.C. Brown and A. Xia, Stein’s method and birth–death processes, Ann. Probab., 29(3):1373–1403, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Čekanavičius and B. Roos, An expansion in the exponent for compound binomial approximation, Lith. Math. J., 46(1):54–91, 2006.

    Article  MATH  Google Scholar 

  8. L.H.Y. Chen, Poisson approximation for dependent trials, Ann. Probab., 3(3):534–545, 1975.

    Article  MATH  Google Scholar 

  9. P. Deheuvels and D. Pfeifer, On a relationship between Uspensky’s theorem and Poisson approximations, Ann. Probab., 40(4):671–681, 1988.

    MathSciNet  MATH  Google Scholar 

  10. J. Dhaene and N. De Pril, On a class of approximative computation methods in individual risk model, Insur. Math. Econ., 14(2):181–196, 1994.

    Article  MATH  Google Scholar 

  11. P. Eichelsbacher and M. Roos, Compound Poisson approximation for dissociated random variables via Stein’s method, Comb. Probab. Comput., 8:335–346, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Heinrich, A method for the derivation of limit theorems for sums of m-dependent random variables, Z. Wahrscheinlichkeitstheor. Verw. Geb., 60:501–515, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Heinrich, A method for the derivation of limit theorems for sums of weakly dependent random variables: A survey, Optimization, 18(5):715–735, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Hipp, Improved approximations for the aggregate claims distribution in the individual model, Astin Bull., 16(2):89–100, 1986.

    Article  MathSciNet  Google Scholar 

  15. J. Kruopis, Precision of approximation of the generalized binomial distribution by convolutions of Poisson measures, Lith. Math. J., 26(1):37–49, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  16. A.M. Mood, The distribution theory of runs, Ann. Math. Stat., 11(4):367–392, 1940.

    Article  MathSciNet  MATH  Google Scholar 

  17. E.L. Presman, Approximation of binomial distributions by infinitely divisible ones, Theory Probab. Appl., 28(2):393–403, 1984.

    Article  MATH  Google Scholar 

  18. E.L. Presman, 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 

  19. A. Röllin, Approximation of sums of conditionally independent variables by the translated Poisson distribution, Bernoulli, 11(6):417–422, 2005.

    Article  Google Scholar 

  20. B. Roos, Asymptotic and sharp bounds in the Poisson approximation to the Poisson-binomial distributions, Bernoulli, 5(6):1021–1034, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Roos, Sharp constants in the Poisson approximation, Stat. Probab. Lett., 52(2):155–168, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Roos, Poisson approximation via the convolution with Kornya–Presman signed measures, Theory Probab. Appl., 48(3):555–560, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  23. R.J. Serfling, A general Poisson approximation theorem, Ann. Probab., 3(4):726–731, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  24. V.V. Shergin, On the convergence rate in the central limit theorem for m-dependent random variables, Theory Probab. Appl., 24(4):782–796, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  25. V. Statulevičius, Large-deviation theorems for sums of dependent random variables I, Lith. Math. J., 19(2):289–296, 1979.

    Article  Google Scholar 

  26. J. Sunklodas, Rate of convergence in the central limit theorem for random variables with strong mixing, Lith. Math. J., 24(2):182–190, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Sunklodas, On a lower bound of the rate of convergence in the central limit theorem for m-dependent random variables, Lith. Math. J., 37(3):291–299, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  28. A.N. Tichomirov, On the convergence rate in the central limit theorem for weakly dependent random variables, Teor. Veroyatn. Primen., 25:800–809, 1980.

    Google Scholar 

  29. Xiaoxin Wang and A. Xia, On negative binomial approximation to k-runs, J. Appl. Probab., 45(2):456–471, 2008.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Vilnius University, Naugarduko 24, LT-03225, Vilnius, Lithuania

    J. Petrauskienė & V. Čekanavičius

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  1. J. Petrauskienė
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  2. V. Čekanavičius
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Correspondence to J. Petrauskienė.

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Petrauskienė, J., Čekanavičius, V. Compound Poisson approximations for sums of 1-dependent random variables. I. Lith Math J 50, 323–336 (2010). https://doi.org/10.1007/s10986-010-9089-x

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