Abstract
Since the appearance of Robbins’s paper (1948) the central limit theorem for a sum of a random number of independent identically distributed random variables is one of the most fundamental results in probability, and explains the appearance of the normal distribution in a whole host of diverse applications in mathematics, physics, biology and the social sciences. Compound random sums are extensions of classical random sums when the numbers of independent summands in sums are partial sums of independent identically distributed positive integer-valued random variables, assumed independent of summands of sums. The main aim of this paper is to introduce central limit theorems for normalized compound random sums of independent random variables and establish the convergence rates in types of small-o and large-\(\mathcal{O}\) estimates, in term of Trotter-distance. The obtained results in this paper are extensions of several known ones.
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REFERENCES
P. L. Butzer, L. Hahn, and U. Westphal, ‘‘On the rate of approximation in the central limit theorem,’’ J. Approx. Theory 13, 327–340 (1975).
P. L. Butzer and L. Hahn, ‘‘General theorems on rates of convergence in distribution of random variables. I. General limit theorems,’’ J. Multivar. Anal. 8, 181–201 (1978).
P. L. Butzer and L. Hahn, ‘‘General theorems on rates of convergence in distribution of random variables. II. Applications to the stable limit laws and weak laws of large numbers,’’ J. Multivar. Anal. 8, 202–221 (1978).
P. L. Butzer and D. Schulz, ‘‘The random martingale central limit theorem and weak law of large numbers with \(o-\) rates,’’ Acta Sci. Math. 45, 81–94 (1983).
L. H. Chen, L. Goldstein, and Q.-M. Shao, Normal Approximation by Stein’s Method, Probability and its Applications (Springer, Heidelberg, 2011).
R. Cioczek and D. Szynal, ‘‘On the convergence rate in terms of the Trotter operator in the central limit theorem without moment conditions,’’ Bull. Polish Acad. Sci. Math. 35, 617–627 (1987).
W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed. (Wiley, New York, 1971), Vol 2.
B. Gnedenko and V. Yu. Korolev, Random Summations: Limit Theorems and Applications (CRC, New York, 1996).
A. Gut, Probability: A Graduate Course, 2nd ed. (Springer, Berlin, 2013).
T. L. Hung, ‘‘On a probability metric based on Trotter operator,’’ Vietnam J. Math. 35, 21–32 (2007).
T. L. Hung, ‘‘Estimations of the Trotter’s distance of two weighted random sums of d-dimensional independent random variables,’’ Int. Math. Forum 4, 1079–1089 (2009).
T. L. Hung and T. T. Thanh, ‘‘Some results on asymptotic behaviors of random sums of independent identically distributed random variables,’’ Commun. Korean Math. Soc. 25, 119–128 (2010).
T. L. Hung and T. T. Thanh, ‘‘On the rate of convergence in limit theorems for random sums via Trotter–distance,’’ J. Inequal. Appl., No. 1, 404 (2013). http://www.journalofinequalitiesandapplications.com/content/2013/1/404.
L. T. Giang and T. L. Hung, ‘‘An extension of random summations of independent and identically distributed random variables,’’ Commun. Korean Math. Soc. 33, 605–618 (2018).
T. L. Hung, ‘‘A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums,’’ Math. Slov. 70, 213–232 (2020). https://doi.org/10.1515/ms-2017-0345
Ü Işlak, ‘‘Asymptotic results for random sums of dependent random variables,’’ Stat. Probab. Lett. 109, 22–29 (2016).
V. Kalashnikov, Geometric Sums: Bounds for Rare Events with Applications (Kluwer Academic, Dordrecht, 1997).
H. Kirschfink, ‘‘The generalized Trotter operator and weak convergence of dependent random variables in different probability metrics,’’ Results Math. 15, 294–323 (1989).
V. M. Kruglov and V. Yu. Korolev, Limit Theorems for Random Sums (Mosk. Gos. Univ., Moscow, 1990) [in Russian].
A. I. Khuri, Advanced Calculus with Applications in Statistics, 2nd ed. (Wiley, Gainesville, FL, 2003).
V. V. Petrov, Limit Theorems of Probability Theory. Sequences of Independent Random Variables (Clarendon, Oxford, 1995).
P. B. L. S. Rao, ‘‘On the rate of approximation in the multidimensional central limit theorem,’’ Liet. Matem. Rink 17, 187–194 (1977).
A. Rényi, Probability Theory (Akademiai Kiado, Budapest, 1970).
H. Robbins, ‘‘The asymptotic distribution of the sums of a random number of random variables,’’ Bull. Am. Math. Soc. 54, 1151–1161 (1948).
Z. Rychlik and D. Szynal, ‘‘Convergence rates in the central limit theorem for sums of a random number of independent random variables,’’ Probab. Theory Appl. 20, 359–370 (1975).
Z. Rychlik and D. Szynal, ‘‘On the rate of convergence in the central limit theorem,’’ Banach Center Publ. 5, 221–229 (1979).
Z. Rychlik and D. Szynal, ‘‘On the rate of approximation in the random-sum central limit theorem,’’ Teor. Veroyatn. Primen. 24, 614–620 (1979).
Z. Rychlik, ‘‘A central limit theorem for sums of a random number of independent random variables,’’ Colloq. Math. 35, 147–158 (1976).
V. Sakalauskas, ‘‘On an estimate in the multidimensional limit theorems,’’ Liet. Matem. Rink. 17, 195–201 (1977).
J. G. Shanthikumar and U. Sumita, ‘‘A central limit theorem for random sums of random variables,’’ Operat. Res. Lett. 3, 153–155 (1984).
P. K. Sen and J. M. Singer, Large Sample Methods in Statistics. An Introduction with Applications (Chapman and Hall, New York, 1993).
H. F. Trotter, ‘‘An elementary proof of the central limit theorem,’’ Arch. Math. (Basel) 10, 226–234 (1959).
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Loc Hung, T. On the Rates of Convergence in Central Limit Theorems for Compound Random Sums of Independent Random Variables. Lobachevskii J Math 42, 374–393 (2021). https://doi.org/10.1134/S1995080221020128
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DOI: https://doi.org/10.1134/S1995080221020128