A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums
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Tran Loc Hung
Abstract
The purpose of this paper is to study a chi-square-type distribution who degrees of freedom are geometric random variables in connection with weak limiting distributions of geometric random sums of squares of independent, standard normal distributed random variables. Some characteristics of chi-square-type random variables with geometrically distributed degrees of freedom including probability density function, probability distribution function, mean and variance are calculated. Some asymptotic behaviors of chi-square-type random variables with geometrically distributed degrees of freedom are also established via weak limit theorems for normalized geometric random sums of squares of independent, standard normal distributed random variables. The rates of convergence in desired weak limit theorems also estimated through Trotter’s distance. The received results are extensions and generalizations of several known results.
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Communicated by Gejza Wimmer
6 Appendix
We must recall that Proposition 2.1 in Section 2 could be directly proved as follows:
Let X be a random variable with 𝔼(|X|k) < +∞. Then 𝔼(|X|j) < +∞, for any 1 ≤ j ≤ k, and
Proof
We shall begin with showing that
Consider the first term of right-side of (6.1), since |X| < 1 for j ≥ 1 and 𝕀(|X| ≤ 1) ≤ 1, it follows that
Hence
According to second term of right-side of (6.1), since |X| > 1 and 1 ≤ j ≤ k, we have
Thus, for 1 ≤ j ≤ k,
Moreover, it follows that
Finally, from (6.1), for each random variable X, if 1 ≤ j ≤ k, it may be concluded that
From above inequality, if 𝔼|X|k < +∞, then 𝔼|X|j < +∞, for 1 ≤ j ≤ k. The proof is complete. □
References
[1] Asmussen, S.: Applied Probability and Queues, Springer, 2003.Search in Google Scholar
[2] Asmussen, S.: Ruin Probabilities, World Scientific, 2010.10.1142/7431Search in Google Scholar
[3] Bon, J. L.: Exponential Approximations of Geometric Sums for Reliability, University of Sciences at Technologies, 2002.Search in Google Scholar
[4] Bon, J. L.: Geometric Sums in reliability Evaluation of Regenerative Systems, Kalashnikov Memorial Seminar (2002), 161–163.Search in Google Scholar
[5] Brown, M.: Error bounds for exponential approximation of geometric convolutions, Ann. Probab. 18(3) (1990), 1388–1402.10.1214/aop/1176990750Search in Google Scholar
[6] Butzer, P. L.—Hahn, L.—Westphal, U.: On the rate of approximation in the central limit theorem, J. Approx. Theory 13 (1975), 327–340.10.1016/0021-9045(75)90042-8Search in Google Scholar
[7] Butzer, P. L.—Hahn, L.: General theorems on rates of convergence in distribution of random variables I. General limit theorems, J. Multivariate Anal. 8 (1978), 181–201.10.1016/0047-259X(78)90071-4Search in Google Scholar
[8] Butzer, P. L. et al.: An extension of the Lindeberg–Trotter operator-theoretic approach to limit theorems for dependent random variables, Acta Sci. Math. 51 (1987), 423–433.Search in Google Scholar
[9] Chen, L. H. Y.—Röllin, A.: Stein couplings for normal approximation, arXiv:1003.6039.Search in Google Scholar
[10] Daly, F.: Compound geometric approximation under a failure rate constraint, J. Appl. Prob. 53 (2016), 700–714.10.1017/jpr.2016.35Search in Google Scholar
[11] Feiueson, A. H.—DeZaney, F. C.: The distribution and properties of a weighted sum of chi squares, National Aeronautics and Space Administration, Washington D. C., May 1968, 1–44.Search in Google Scholar
[12] Gamboa, F.: Random sums stopped by a rare event: a new approximation, Probab. Math. Statist. 24(2) (2004), 237–252.Search in Google Scholar
[13] Gnedenko, B.: Limit theorems for sums of a random number of positive independent random variables, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, Univ. California Press, Berkeley, Calif., 1972, pp. 537–549.Search in Google Scholar
[14] Gnedenko, B.: On limit theorems for a random number of random variables, Probability Theory and Mathematical Statistics (Tbilisi, 1982), Lecture Notes in Math. 1021, Springer, Berlin, 1983, pp. 167–176.Search in Google Scholar
[15] Gnedenko, B.—Korolev, B.: Random Summation. Limit Theorems and Applications, CRC Press, New York, 1996.Search in Google Scholar
[16] Gordienko, E. I.: Stability estimates of generalized geometric sums and their applications, Kybernetika 40(2) (2004), 257–272.Search in Google Scholar
[17] Grandell, J.: Risk theory and geometric sums, Information Processes 2(2) (2002), 180–181.Search in Google Scholar
[18] Gut, A.: Probability: a Graduate Course, Springer Texts in Statistics, Springer, New York, 2005.Search in Google Scholar
[19] Hogg, R. V. et al.: Introduction to Mathematical Statistics, Seventh Edition, Pearson, 2013.Search in Google Scholar
[20] Hung, T. L. et al.: Some results related to distribution functions of chi-square-type with random degrees of freedom, Bull. Korean Math. Soc. 45(3) (2008), 509–522.10.4134/BKMS.2008.45.3.509Search in Google Scholar
[21] Hung, T. L.: On a probability metric based on Trotter operator. Vietnam J. Math. 35(1) (2007), 21–32.Search in Google Scholar
[22] Hung, T. L.: On the rate of convergence in limit theorems for geometric sums. Southeast Asian J. Sci. 2(2) (2013), 117–130.10.1186/1029-242X-2013-404Search in Google Scholar
[23] Hung, T. L.—Hau, T. N.: On the accuracy of approximation of the distribution of negative-binomial random sums by the gamma distribution, Kybernetika 2(5) (2018), 921–936.10.14736/kyb-2018-5-0921Search in Google Scholar
[24] Hung, T. L.: On chi-square type distributions with geometric degrees of freedom in relation to geometric sums, Science & Technology Development Journal 22(1) (2019), 1–5.10.32508/stdj.v22i1.1053Search in Google Scholar
[25] Kalashnikov, V.: Geometric Sums: Bounds for Rare Events with Applications. Risk Analysis, Reliability, Queueing. Mathematics and its Applications 413, Kluwer, Dordrecht, 1997.10.1007/978-94-017-1693-2Search in Google Scholar
[26] Kirschfink, H.: The generalized Trotter operator and weak convergence of dependent random variables in different probability metrics, Results Math. 15 (1989), 294–323.10.1007/BF03322619Search in Google Scholar
[27] Klebanov, L. B. et al.: A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables, Theory Probab. Appl. 29 (1984), 791–794.10.1137/1129104Search in Google Scholar
[28] Klebanov, L. B.: Heavy Tailed Distribution, Prague, 2003.Search in Google Scholar
[29] Kruglov, V. M.—Korolev, V. Yu.: Limit Theorems for Random Sums, Moscow University Press, Moscow, 1990, (in Russian).Search in Google Scholar
[30] Kotz, S. et al.: The Laplace Distribution and Generalization, Springer Science + Business Media, LLC., 2001.10.1007/978-1-4612-0173-1Search in Google Scholar
[31] Mawaziny, A. H.: Chi-square Distribution Theory with Applications to Reliability Problems. Retrospective Theses and Dissertations 4037, 1965, http://lib.dr.iastate.edu/rtd.10.31274/rtd-180817-4404Search in Google Scholar
[32] Peköz, E. A.—Röllin, A.: New rates for exponential approximation and the theorems of Rényi and Yaglom, Ann. Probab. 39(2) (2011), 587–608.10.1214/10-AOP559Search in Google Scholar
[33] Petrov, V. V.: Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Clarendom Press, Oxford, 1995.Search in Google Scholar
[34] Pike, J.—Ren, H.: Stein’s method and the Laplace distribution, ALEA Lat. Am. J. Probab. Math. Stat. 11(2) (2014), 571–587.Search in Google Scholar
[35] Rahman, G. et al.: Generalization of Chi-square distribution, Journal of Statistics Applications and Probability 4(1) (2015), 119–126.Search in Google Scholar
[36] Rényi, A.: On the central limit theorem for the sum of a random number of independent random variables, Acta Math. Hungar. 11(1–2) (1960), 97–102.10.1007/BF02020627Search in Google Scholar
[37] Sunklodas, J. K.: On the normal approximation of a negative binomial random sum, Lith. Math. J. 55(1) (2015), 150–158.10.1007/s10986-015-9271-2Search in Google Scholar
[38] Trotter, H. F.: An elementary proof of the central limit theorem, Arch. Math. 10 (1959), 226–234.10.1007/BF01240790Search in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- Doc. RNDr. Roman Frič, DrSc. – 3/4 C?
- Sugihara algebras and Sugihara monoids: Multisorted dualities
- Two by two squares in set partitions
- Identifying quantum logics by numerical events
- Some remarks on divisible polyhedral MV-algebras
- Remarks on b-metrics, ultrametrics, and metric-preserving functions
- Estimates of essential norm of generalized weighted composition operators from Bloch type spaces to nth weighted type spaces
- Maximum term of transcendental entire function and spider’s web
- Entire solutions of certain type of non-linear differential equations
- On the oscillatory behavior of third order differential equations with a sublinear neutral term
- Eigenvalue problems for a class of nonlinear Hadamard fractional differential equations with p-Laplacian operator
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The amalgam space
- On the topology of partial metric spaces
- The trace of the commutator of the Cesàro operator and a compact operator
- Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold
- On 4-th root metrics of isotropic scalar curvature
- On ultrametric-preserving functions
- Porosity of 𝓢-approximately continuous functions
- On a modified Burr XII distribution having flexible hazard rate shapes
- A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums
- An alternative estimate for the numerical radius of Hilbert space operators
