Abstract
For a Markov binomial distribution, we prove partial cases of the uniform Kolmogorov and Simons–Johnsontheorems. We show that, in total variation, the accuracy of approximation by the class of all infinitely divisible distributions is of the order O(n −2/3). We also prove the convergence to a compound Poisson distribution with exponential weights.
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T.V. Arak and A.Yu. Za˘ıtsev, Uniform limit theorems for sums of independent random variables, Proc. Steklov Inst. Math., 175:1–222, 1988.
A.D. Barbour, L.H.Y. Chen, and K.P. Choi, Poisson approximation for unbounded functions, I: Independent summands, Stat. Sinica, 5:749–766, 1995.
A.D. Barbour and V. Čekanavičius, Total variation asymptotics for sums of independent integer random variables, Ann. Probab., 30:509–545, 2002.
I. S. Borisov and P. S. Ruzankin, Poisson approximation for expectations of unbounded functions of independent random variables, Ann. Probab., 30:1657–1680, 2002.
L.H.Y. Chen and M. Roos, Compound Poisson approximation for unbounded functions on a group with application to large deviations, Probab. Theory Related Fields, 103:515–528, 1995.
R.L. Dobrushin, Limit theorems for a Markov chain of two states, Izv. Akad. Nauk SSSR, Ser. Mat., 17:291–330, 1953. English transl.: Sel. Transl. Math. Stat. Probab., 1:97–134 1961.
J. Gani, On the probability generating function of the sum of Markov–Bernoulli random variables, Essays in statistical science: Papers in Honour of P.A.P. Moran, J. Appl. Probab., Spec. Vol., 19A:321–326, 1982.
S.R. Hsiau, Compound Poisson limit theorems for Markov chains, J. Appl. Probab., 34:24–34, 1997.
A.N. Kolmogorov, Two uniform limit theorems for sums of independent random variables, Theory Probab. Appl., 1:384–394, 1956.
É.L. Presman, Approximation of binomial distributions by infinitely divisible ones, Theory Probab. Appl., 28:393–403, 1983.
É.L. Presman, Approximation in variation of the distribution of a sum of independent Bernoulli variables with a Poisson law, Theory Probab. Appl., 30:417–422, 1986.
R.J. Serfling, A general Poisson approximation theorem, Ann. Probab., 3:726–731, 1975.
G. Simons and N.L. Johnson, On the convergence of binomial to Poisson distributions, Ann. Math. Stat., 42:1735–1736, 1971.
V. Čekanavičius and P. Vellaisamy, Compound Poisson and signed compound Poisson approximations to theMarkov binomial law, Bernoulli, 16(4):1114–1136, 2010.
V. Čekanavičius and P. Vellaisamy, A compound Poisson convergence theorem for sums of m-dependent variables, J. Theor. Probab., 2014, doi:10.1007/s10959-014-0540-5.
V. Čekanavičius and M. Mikalauskas, Local theorems for the Markov binomial distribution, Lith. Math. J., 41:219–231, 2001.
V. Čekanavičius , On the convergence of Markov binomial to Poisson distribution, Stat. Probab. Lett., 58:83–91, 2002.
P. Vellaisamy and B. Chaudhuri, On compound Poisson approximation for sums of random variables, Stat. Probab. Lett., 41:179–189, 1999.
Y.H. Wang, A compound Poisson convergence theorem, Ann. Probab., 19:452–455, 1991.
Y.H. Wang, Approximating kth-order two-state Markov chains, J. Appl. Probab., 29:861–868, 1992.
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Šliogere, J., Čekanavičius, V. Two limit theorems for Markov binomial distribution. Lith Math J 55, 451–463 (2015). https://doi.org/10.1007/s10986-015-9291-y
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DOI: https://doi.org/10.1007/s10986-015-9291-y