Mynbayev, Kairat and Darkenbayeva, Gulsim
(2019):
*Analyzing variance in central limit theorems.*
Published in: Kazakh Mathematical Journal
, Vol. 19, No. 3
(2019): pp. 30-39.

Preview |
PDF
MPRA_paper_101685.pdf Download (401kB) | Preview |

## Abstract

Central limit theorems deal with convergence in distribution of sums of random variables. The usual approach is to normalize the sums to have variance equal to 1. As a result, the limit distribution has variance one. In most papers, existence of the limit of the normalizing factor is postulated and the limit itself is not studied. Here we review some results which focus on the study of the normalizing factor. Applications are indicated.

Item Type: | MPRA Paper |
---|---|

Original Title: | Analyzing variance in central limit theorems |

Language: | English |

Keywords: | Central limit theorems, convergence in distribution, limit distribution, variance |

Subjects: | C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C10 - General C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C40 - General |

Item ID: | 101685 |

Depositing User: | Kairat Mynbaev |

Date Deposited: | 19 Jul 2020 01:51 |

Last Modified: | 19 Jul 2020 01:51 |

References: | 1. Lindeberg, J. W. (1922). "Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung". Mathematische Zeitschrift. 15 (1): 211-225 2. Davidson, J. 1994. Stochastic Limit Theory: An introduction for econometricians. New York: Oxford University Press. 3. W. Hoeffding and H. Robbins, The central limit theorem for dependent random variables, Duke Mathematical Journal, vol. 15, pp. 773-780, 1948. 4. P. H. Diananda, The central limit theorem for m-dependent variables,vol. 51, pp. 92-95, 1955. 5. K. N. Berk, A central limit theorem for m-dependent random variables with unbounded m, Annals of Probability, vol. 1, no. 2, pp. 352-354, 1973. 6. M. Rosenblatt, A central limit theorem and a strong mixing condition, Proceedings of the National Academy of Sciences of the United States of America, vol. 42, no. 1, pp. 43-47, 1956. 7. Ibragimov, I. A., 1962. Some limit theorems for stationary processes, Theory of Probability and its Applications, 7, 349-82. 8. Eicker, F. 1966. A multivariate central limit theorem for random linear vector forms. Ann. Math. Stat., 37, 1825-1828. 9. Serfling, R. J. (1968). Contributions to central limit theory for dependent variables. Ann. Math. Statist. 39 1158-1175. 10. Gordin, M.I.: The central limit theorem for stationary processes. Soviet Math. Dokl. 10, 1174-1176 (1969) 11. Gordin, M.I. (2004). A remark on the martingale method for proving the central limit theorem for stationary sequences. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI) 311. Veroyatn. I Stat. 7 124--132, 299--300. Transl.: J. Math. Sci. (N.Y.) 133 (2006) 1277-1281. 12. Dvoretzky, A. 1972. Asymptotic normality for sums of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory. pp. 513-535, Berkeley: Univ. California Press. 13. McLeish, D. L. 1974. Dependent central limit theorems and invariance principles. Ann. Prob., 2, 620-628. 14. Hannan, E. J., 1979. "The central limit theorem for time series regression," Stochastic Processes and their Applications, Elsevier, vol. 9(3), pages 281-289. 15. Hahn, M. G., Kuelbs, J., Samur, J. D. 1987. Asymptotic normality of trimmed sums of mixing random variables. Ann. Probab., 15, 1395-1418. 16. De Jong, R. M., 1997. Central limit theorems for dependent heterogeneous random variables. Econometric Theory 13, 353-67. 17. Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713-724. 18. Heyde, C.C. (1974). On the central limit theorem for stationary processes. Z. Wahrsch. Verw. Gebiete. 30 315-320. 19. Heyde, C.C.: On the central limit theorem and iterated logarithm law for stationary processes. Bull. Austral. Math. Soc. Volume 12, 1975, pp. 1-8. 20. T. C. Christofides and P. M. Mavrikiou, Central limit theorem for dependent multidimensionally indexed random variables, Statistics & Probability Letters, vol. 63, no. 1, pp. 67-78, 2003. 21. M. Kaminski, Central limit theorem for certain classes of dependent random variables, Theory of Probability and its Applications, vol. 51, no. 2, pp. 335-342, 2007. 22. Y. Shang, A central limit theorem for randomly indexed m-dependent random variables, Filomat, vol. 26, no. 4, pp. 713-717, 2012. 23. Dedecker, J. and Merlev`ede, F. (2002). Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 1044-1081. 24. Davidson, J., 1992. A central limit theorem for globally nonstationary near-epoch dependent functions of mixing processes, Econometric Theory 8, 313-29. 25. Davidson, J., 1993. The central limit theorem for globally non-stationary near-epoch dependent functions of mixing processes: the asymptotically degenerate case, Econometric Theory 9, 402-12. 26. Avram, F. and Fox, R. (1992). Central limit theorems for sums of Wick products of stationary sequences. Trans. Amer. Math. Soc. 330 651-663. 27. Giraitis, L. and Taqqu, M. S. (1997). Limit theorems for bivariate Appell polynomials. I. Central limit theorems. Probab. Theory Related Fields 107, 359-381. 28. Ho, H. C. and Sun, T. C. (1987). A central limit theorem for non-instantaneous filters of a stationary Gaussian process. J. Multivariate Anal. 22 144-155 29. M. Peligrad, S. Utev, Central limit theorem for stationary linear processes, Ann. Probab. 34(4) (2006) 1608-1622. 30. K. Mynbaev, Lp-approximable sequences of vectors and limit distribution of quadratic forms of random variables, Adv. in Appl. Math. 26(4) (2001) 302-329. 31. K. Mynbaev, Central limit theorems for weighted sums of linear processes: Lp-approximability versus Brownian motion, Econometric Theory 25(3) (2009) 748-763. 32. K. Mynbaev, A. Ullah, Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model, J. Multivariate Anal. 99(2) (2008) 245-277. 33. Mynbaev, K. T. 2010. Asymptotic distribution of the OLS estimator for a mixed regressive, spatial autoregressive model. J. Multivar. Anal., 10(3), 733-748. 34. Mynbaev, K. T. 2011. Regressions with asymptotically collinear regressors. Vol. 14, No. 2 (2011), pp. 304-320. 35. K.T. Mynbaev, G.S. Darkenbayeva. Weak convergence of linear and quadratic forms and related statements on Lp-approximability. J. Math. Anal. Appl. 473 (2019) 1305-1319. 37. K. Mynbaev, Short-Memory Linear Processes and Econometric Applications, Wiley and Sons, 2011. 38. S. Nabeya, K. Tanaka, Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative, Ann. Statist. 16(1) (1988) 218-235. 39. K. Tanaka, Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Wiley and Sons, 1996. 40. W. Wu, X. Shao, Asymptotic spectral theory for nonlinear time series, Ann. Statist. 35(4) (2007) 1773-1801. 41. Phillips, P. C. B. 2007. Regression with slowly varying regressors and nonlinear trends. Economet. Theor., 23, 557-614. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/101685 |