Kazakh Mathematical Journal. 2019; : 30-39
Анализ дисперсии в центральных предельных теоремах
Аннотация
Центральные предельные теоремы связаны со сходимостью по распределению сумм случайных величин. Обычный подход заключается в нормализации сумм так, чтобы иметь дисперсию, равную единице. В результате этого предельное распределение имеет дисперсию, равную единице. Во многих работах существование предела нормализующего фактора постулируется, а сам предел не изучен. Здесь мы рассмотрим некоторые результаты, которые сосредоточены на изучении коэффициента нормализации. Указаны их области применения.
Список литературы
1. Lindeberg J.W. Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift, 15:1 (1922), 211-225. https://doi.org/10.1007/BF01494395.
2. Davidson J. Stochastic Limit Theory: An introduction for econometricians, New York: Oxford University Press, 1994.
3. Hoeffding W., Robbins H. The central limit theorem for dependent random variables, Duke Mathematical Journal, 15 (1948), 773-780. https://doi.org/10.1215/S0012-7094-48-01568-3.
4. Diananda P.H. The central limit theorem for m-dependent variables, Mathematical Proceedings of the Cambridge Philosophical Society kiskashasy, 51 (1955), 92-95. https://doi.org/10.1017/S0305004100029959.
5. Berk K.N. A central limit theorem for m-dependent random variables with unbounded m, Annals of Probability, 1:1 (1973), 352-354.
6. Rosenblatt M. A central limit theorem and a strong mixing condition, Proceedings of the National Academy of Sciences of the United States of America, 42:1 (1956), 43-47. https://doi.org/10.1073/pnas.42.1.43.
7. Ibragimov I.A. Some limit theorems for stationary processes, Theory of Probability and its Applications, 7 (1962), 349-382. https://doi.org/10.1137/1107036.
8. Eicker F. A multivariate central limit theorem for random linear vector forms, Ann. Math. Stat., 37 (1966), 1825-1828. https://doi.org/10.1214/aoms/1177699175.
9. Serfling R.J. Contributions to central limit theory for dependent variables, Ann. Math. Statist., 39 (1968), 1158-1175. https://doi.org/10.1214/aoms/1177698240.
10. Gordin M.I. The central limit theorem for stationary processes, Soviet Math. Dokl., 10 (1969), 1174–176.
11. Gordin M.I. 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 (2004), 124-132, 299-300. Transl.: J. Math. Sci. (N.Y.), 133 (2006) 1277-1281.
12. Dvoretzky A. 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, 1972, 513-535, Berkeley: Univ. California Press.
13. McLeish D.L. Dependent central limit theorems and invariance principles, Ann. Prob., 2 (1974), 620-628. https://doi.org/10.1214/aop/1176996608.
14. Hannan E.J. The central limit theorem for time series regression, Stochastic Processes and their Applications, Elsevier, 9:3 (1979), 281-289. https://doi.org/10.1016/0304-4149(79)90050-4.
15. Hahn M.G., Kuelbs J., Samur J.D. Asymptotic normality of trimmed sums of mixing random variables, Ann. Probab., 15 (1987), 1395-1418. https://doi.org/10.1214/aop/1176991984.
16. De Jong R.M. Central limit theorems for dependent heterogeneous random variables, Econometric Theory, 13 (1997), 353-367. https://doi.org/10.1017/S0266466600005843.
17. Maxwell M., Woodroofe M. Central limit theorems for additive functionals of Markov chains, Ann. Probab., 28 (2000), 713-724. https://doi.org/10.1214/aop/1019160258.
18. Heyde C.C. On the central limit theorem for stationary processes, Z. Wahrsch. Verw. Gebiete, 30 (1974), 315-320. https://doi.org/10.1007/BF00532619.
19. Heyde C.C. On the central limit theorem and iterated logarithm law for stationary processes, Bull. Austral. Math. Soc., 12 (1975), 1-8.
20. Christofides T.C., Mavrikiou P.M. Central limit theorem for dependent multidimensionally indexed random variables, Statistics & Probability Letters, 63:1 (2003), 67-78.
21. Kaminski M. Central limit theorem for certain classes of dependent random variables, Theory of Probability and its Applications, 51:2 (2007), 335-342. https://doi.org/10.4213/tvp65.
22. Shang Y. A central limit theorem for randomly indexed m-dependent random variables, Filomat, 26:4 (2012), 713-717. https://doi.org/10.2298/FIL1204713S.
23. Dedecker J., Merlev‘ede F. Necessary and sufficient conditions for the conditional central limit theorem, Ann. Probab., 30 (2002), 1044-1081. https://www.jstor.org/stable/1558793.
24. Davidson J. A central limit theorem for globally nonstationary near-epoch dependent functions of mixing processes, Econometric Theory, 8 (1992), 313-329. https://doi.org/10.1017/S0266466600012950.
25. Davidson J. The central limit theorem for globally non-stationary near-epoch dependent functions of mixing processes: the asymptotically degenerate case, Econometric Theory, 9 (1993), 402-412. https://doi.org/10.1017/S0266466600007738.
26. Avram F., Fox R. Central limit theorems for sums of Wick products of stationary sequences, Trans. Amer. Math. Soc., 330 (1992), 651-663. https://doi.org/10.2307/2153927.
27. Giraitis L., Taqqu M.S. Limit theorems for bivariate Appell polynomials. I. Central limit theorems, Probab. Theory Related Fields, 107 (1997), 359-381.
28. Ho H.C., Sun T.C. A central limit theorem for non-instantaneous filters of a stationary Gaussian process, J. Multivariate Anal., 22 (1987), 144-155. https://doi.org/10.1016/0047-259X(87)90082-0.
29. Peligrad M., Utev S. Central limit theorem for stationary linear processes, Ann. Probab., 34:4 (2006), 1608-1622. https://doi.org/10.1214/009117906000000179.
30. Mynbaev K.T. Lp-approximable sequences of vectors and limit distribution of quadratic forms of random variables, Adv. in Appl. Math., 26:4 (2001), 302-329. https://doi.org/10.1006/aama.2001.0723.
31. Mynbaev K.T. Central limit theorems for weighted sums of linear processes: Lp-approximability versus Brownian motion, Econometric Theory, 25:3 (2009), 748-763. https://doi.org/10.1017/S0266466608090282.
32. Mynbaev K.T., Darkenbayeva G.S. Weak convergence of linear and quadratic forms and related statements on Lp-approximability, J. Math. Anal. Appl., 473 (2019), 1305-1319. https://doi.org/10.1016/j.jmaa.2019.01.023.
33. Nabeya S., Tanaka K. Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative, Ann. Statist., 16:1 (1988), 218-235. https://doi.org/10.1214/aos/1176350701.
34. Tanaka K. Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Wiley and Sons, 1996.
35. Wu W., Shao X. Asymptotic spectral theory for nonlinear time series, Ann. Statist., 35:4 (2007), 1773-1801. https://doi.org/10.1214/009053606000001479.
36. Phillips P.C.B. Regression with slowly varying regressors and nonlinear trends, Economet. Theor., 23 (2007), 557-614. https://doi.org/10.1017/S0266466607070260.
37. Mynbaev K.T. Regressions with asymptotically collinear regressors, The Econometrics Journal, 14:2 (2011), 304-320. https://doi.org/10.1111/j.1368-423X.2010.00334.x.
38. Mynbaev K.T., Ullah A. Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model, J. Multivariate Anal., 99:2 (2008), 245-277. https://doi.org/10.1016/j.jmva.2007.04.002.
39. Mynbaev K.T. Asymptotic distribution of the OLS estimator for a mixed regressive, spatial autoregressive model, J. Multivar. Anal., 10:3 (2010), 733-748. https://doi.org/10.1016/j.jmva.2009.11.003.
40. Mynbaev K.T. Short-Memory Linear Processes and Econometric Applications, Wiley and Sons, 2011.
Kazakh Mathematical Journal. 2019; : 30-39
Analyzing variance in central limit theorems
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.
References
1. Lindeberg J.W. Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift, 15:1 (1922), 211-225. https://doi.org/10.1007/BF01494395.
2. Davidson J. Stochastic Limit Theory: An introduction for econometricians, New York: Oxford University Press, 1994.
3. Hoeffding W., Robbins H. The central limit theorem for dependent random variables, Duke Mathematical Journal, 15 (1948), 773-780. https://doi.org/10.1215/S0012-7094-48-01568-3.
4. Diananda P.H. The central limit theorem for m-dependent variables, Mathematical Proceedings of the Cambridge Philosophical Society kiskashasy, 51 (1955), 92-95. https://doi.org/10.1017/S0305004100029959.
5. Berk K.N. A central limit theorem for m-dependent random variables with unbounded m, Annals of Probability, 1:1 (1973), 352-354.
6. Rosenblatt M. A central limit theorem and a strong mixing condition, Proceedings of the National Academy of Sciences of the United States of America, 42:1 (1956), 43-47. https://doi.org/10.1073/pnas.42.1.43.
7. Ibragimov I.A. Some limit theorems for stationary processes, Theory of Probability and its Applications, 7 (1962), 349-382. https://doi.org/10.1137/1107036.
8. Eicker F. A multivariate central limit theorem for random linear vector forms, Ann. Math. Stat., 37 (1966), 1825-1828. https://doi.org/10.1214/aoms/1177699175.
9. Serfling R.J. Contributions to central limit theory for dependent variables, Ann. Math. Statist., 39 (1968), 1158-1175. https://doi.org/10.1214/aoms/1177698240.
10. Gordin M.I. The central limit theorem for stationary processes, Soviet Math. Dokl., 10 (1969), 1174–176.
11. Gordin M.I. 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 (2004), 124-132, 299-300. Transl.: J. Math. Sci. (N.Y.), 133 (2006) 1277-1281.
12. Dvoretzky A. 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, 1972, 513-535, Berkeley: Univ. California Press.
13. McLeish D.L. Dependent central limit theorems and invariance principles, Ann. Prob., 2 (1974), 620-628. https://doi.org/10.1214/aop/1176996608.
14. Hannan E.J. The central limit theorem for time series regression, Stochastic Processes and their Applications, Elsevier, 9:3 (1979), 281-289. https://doi.org/10.1016/0304-4149(79)90050-4.
15. Hahn M.G., Kuelbs J., Samur J.D. Asymptotic normality of trimmed sums of mixing random variables, Ann. Probab., 15 (1987), 1395-1418. https://doi.org/10.1214/aop/1176991984.
16. De Jong R.M. Central limit theorems for dependent heterogeneous random variables, Econometric Theory, 13 (1997), 353-367. https://doi.org/10.1017/S0266466600005843.
17. Maxwell M., Woodroofe M. Central limit theorems for additive functionals of Markov chains, Ann. Probab., 28 (2000), 713-724. https://doi.org/10.1214/aop/1019160258.
18. Heyde C.C. On the central limit theorem for stationary processes, Z. Wahrsch. Verw. Gebiete, 30 (1974), 315-320. https://doi.org/10.1007/BF00532619.
19. Heyde C.C. On the central limit theorem and iterated logarithm law for stationary processes, Bull. Austral. Math. Soc., 12 (1975), 1-8.
20. Christofides T.C., Mavrikiou P.M. Central limit theorem for dependent multidimensionally indexed random variables, Statistics & Probability Letters, 63:1 (2003), 67-78.
21. Kaminski M. Central limit theorem for certain classes of dependent random variables, Theory of Probability and its Applications, 51:2 (2007), 335-342. https://doi.org/10.4213/tvp65.
22. Shang Y. A central limit theorem for randomly indexed m-dependent random variables, Filomat, 26:4 (2012), 713-717. https://doi.org/10.2298/FIL1204713S.
23. Dedecker J., Merlev‘ede F. Necessary and sufficient conditions for the conditional central limit theorem, Ann. Probab., 30 (2002), 1044-1081. https://www.jstor.org/stable/1558793.
24. Davidson J. A central limit theorem for globally nonstationary near-epoch dependent functions of mixing processes, Econometric Theory, 8 (1992), 313-329. https://doi.org/10.1017/S0266466600012950.
25. Davidson J. The central limit theorem for globally non-stationary near-epoch dependent functions of mixing processes: the asymptotically degenerate case, Econometric Theory, 9 (1993), 402-412. https://doi.org/10.1017/S0266466600007738.
26. Avram F., Fox R. Central limit theorems for sums of Wick products of stationary sequences, Trans. Amer. Math. Soc., 330 (1992), 651-663. https://doi.org/10.2307/2153927.
27. Giraitis L., Taqqu M.S. Limit theorems for bivariate Appell polynomials. I. Central limit theorems, Probab. Theory Related Fields, 107 (1997), 359-381.
28. Ho H.C., Sun T.C. A central limit theorem for non-instantaneous filters of a stationary Gaussian process, J. Multivariate Anal., 22 (1987), 144-155. https://doi.org/10.1016/0047-259X(87)90082-0.
29. Peligrad M., Utev S. Central limit theorem for stationary linear processes, Ann. Probab., 34:4 (2006), 1608-1622. https://doi.org/10.1214/009117906000000179.
30. Mynbaev K.T. Lp-approximable sequences of vectors and limit distribution of quadratic forms of random variables, Adv. in Appl. Math., 26:4 (2001), 302-329. https://doi.org/10.1006/aama.2001.0723.
31. Mynbaev K.T. Central limit theorems for weighted sums of linear processes: Lp-approximability versus Brownian motion, Econometric Theory, 25:3 (2009), 748-763. https://doi.org/10.1017/S0266466608090282.
32. Mynbaev K.T., Darkenbayeva G.S. Weak convergence of linear and quadratic forms and related statements on Lp-approximability, J. Math. Anal. Appl., 473 (2019), 1305-1319. https://doi.org/10.1016/j.jmaa.2019.01.023.
33. Nabeya S., Tanaka K. Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative, Ann. Statist., 16:1 (1988), 218-235. https://doi.org/10.1214/aos/1176350701.
34. Tanaka K. Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Wiley and Sons, 1996.
35. Wu W., Shao X. Asymptotic spectral theory for nonlinear time series, Ann. Statist., 35:4 (2007), 1773-1801. https://doi.org/10.1214/009053606000001479.
36. Phillips P.C.B. Regression with slowly varying regressors and nonlinear trends, Economet. Theor., 23 (2007), 557-614. https://doi.org/10.1017/S0266466607070260.
37. Mynbaev K.T. Regressions with asymptotically collinear regressors, The Econometrics Journal, 14:2 (2011), 304-320. https://doi.org/10.1111/j.1368-423X.2010.00334.x.
38. Mynbaev K.T., Ullah A. Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model, J. Multivariate Anal., 99:2 (2008), 245-277. https://doi.org/10.1016/j.jmva.2007.04.002.
39. Mynbaev K.T. Asymptotic distribution of the OLS estimator for a mixed regressive, spatial autoregressive model, J. Multivar. Anal., 10:3 (2010), 733-748. https://doi.org/10.1016/j.jmva.2009.11.003.
40. Mynbaev K.T. Short-Memory Linear Processes and Econometric Applications, Wiley and Sons, 2011.
