On non-uniform Berry-Esseen bounds for time series

• Autorzy: Jirak M.
• Języki publikacji: EN

Abstrakty

EN

Given a stationary sequence {X_k}_{k ϵ Z}, non-uniform bounds for the normal approximation in the Kolmogorov metric are established. The underlying weak dependence assumption includes many popular linear and nonlinear time series from the literature, such as ARMA or GARCH models. Depending on the number of moments p, typical bounds in this context are of the size O(m^p−1 n^−p/2+1), where we often find that m = m_n = log n. In our setup, we can essentially improve upon this rate by the factor m^−p/2, yielding a bound of O(m^p/2−1 n^−p/2+1). Among other things, this allows us to recover a result from the literature, which is due to Ibragimov.

Słowa kluczowe

EN

Berry-Esseen; weak dependence

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2015
• Tom: Vol. 35, Fasc. 1
• Strony: 1--14
• Opis fizyczny: Bibliogr. 35 poz.

Twórcy

autor

Jirak M.

• Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, D-10099 Berlin, Germany

Bibliografia

• [1] D. W. K. Andrews, Non-strong mixing autoregressive processes, J. Appl. Probab. 21 (1984), pp. 930-934.
• [2] A. Aue, S. Hörmann, L. Horváth, and M. Reimherr, Break detection in the covariance structure of multivariate time series models, Ann. Statist. 37 (2009), pp. 4046-4087.
• [3] V. Bentkus, Asymptotic expansions for distributions of sums of independent random elements in a Hilbert space, Litovsk. Mat. Sb. 4 (1984), pp. 29-48.
• [4] V. Bentkus, F. Götze, and A. Tikhomirov, Berry-Esseen bounds for statistics of weakly dependent samples, Bernoulli 3 (1997), pp. 329-349.
• [5] I. Berkes, S. Hörmann, and L. Horváth, The functional central limit theorem for a family of GARCH observations with applications, Statist. Probab. Lett. 78 (2008), pp. 2725-2730.
• [6] P. Billingsley, Convergence of Probability Measures, Wiley, New York 1968.
• [7] E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab. 10 (1982), pp. 672-688.
• [8] P. Bougerol and N. Picard, Stationarity of GARCH processes and of some nonnegative time series, J. Econometrics 52 (1992), pp. 115-127.
• [9] P. Bougerol and N. Picard, Strict stationarity of generalized autoregressive processes, Ann. Probab. 20 (1992), pp. 1714-1730.
• [10] R. C. Bradley, Introduction to Strong Mixing Conditions, Vol. 1, Kendrick Press, Heber City, UT, 2007.
• [11] L. Chen and Q. Shao, Normal approximation under local dependence, Ann. Probab. 32 (2004), pp. 1985-2028.
• [12] J. Dedecker and P. Doukhan, A new covariance inequality and applications, Stochastic Process. Appl. 106 (2003), pp. 63-80.
• [13] P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev. 41 (1999), pp. 45-76.
• [14] P. Doukhan and O. Wintenberger, An invariance principle for weakly dependent stationary general models, Probab. Math. Statist. 27 (1) (2007), pp. 45-73.
• [15] M. El Machkouri, Berry-Esseen’s central limit theorem for non-causal linear processes in Hilbert space, Afr. Diaspora J. Math. 10 (2) (2010), pp. 81-86.
• [16] M. El Machkouri, D. Volný, and W. B. Wu, A central limit theorem for stationary random fields, Stochastic Process. Appl. 123 (1) (2013), pp. 1-14.
• [17] J. Gao, Nonlinear Time Series: Semiparametric and Nonparametric Methods, Monogr. Statist. Appl. Probab., Vol. 108, Chapman & Hall/CRC, Boca Raton, FL, 2007.
• [18] M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), pp. 739-741.
• [19] S. Hörmann, Berry-Esseen bounds for econometric time series, ALEA Lat. Am. J. Probab. Math. Stat. 6 (2009), pp. 377-397.
• [20] I. Ibragimov, The central limit theorem for sums of functions of independent variables and for sums of the form Σ f(2kt), Theory Probab. Appl. 12 (1967), pp. 596-607.
• [21] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, second edition, Grundlehren Math. Wiss., Vol. 288, Springer, Berlin 2003.
• [22] M. Jirak, Change-point analysis in increasing dimension, J. Multivariate Anal. 111 (2012), pp. 136-159.
• [23] M. Jirak, On weak invariance principles for sums of dependent random functionals, Statist. Probab. Lett. 83 (10) (2013), pp. 2291-2296.
• [24] V. Ladokhin and D. Moskvin, An estimate for the remainder term in the central limit theorem for sums of functions of independent variables and sums of the form Σ f(t2k), Theory Probab. Appl. 16 (1971), pp. 116-125.
• [25] D. L. McLeish, A maximal inequality and dependent strong laws, Ann. Probab. 3 (1975), pp. 829-839.
• [26] W. Min and W. Wu, On linear processes with dependent innovations, Stochastic Process. Appl. 115 (2005), pp. 939-958.
• [27] M. Peligrad and S. Utev, A new maximal inequality and invariance principle for stationary sequences, Ann. Probab. 33 (2005), pp. 798-815.
• [28] M. Peligrad and S. Utev, Central limit theorem for stationary linear processes, Ann. Probab. 34 (2006), pp. 1608-1622.
• [29] M. B. Priestley, Non-linear and Non-stationary Time Series Analysis, Academic Press, London 1988.
• [30] E. Rio, Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes, Probab. Theory Related Fields 104 (1996), pp. 255-282.
• [31] A. Tikhomirov, On the convergence rate in the central limit theorem for weakly dependent random variables, Theory Probab. Appl. 25 (1981), pp. 790-809.
• [32] H. Tong, Non-linear Time Series: A Dynamical System Approach. With an Appendix by K. S. Chan, Oxford Statist. Sci. Ser., Vol. 6, Oxford University Press, New York 1990.
• [33] W. B. Wu, Nonlinear system theory: Another look at dependence, Proc. Natl. Acad. Sci. USA 102 (2005), pp. 14150-14154.
• [34] W. B. Wu, Strong invariance principles for dependent random variables, Ann. Probab. 35 (2007), pp. 2294-2320.
• [35] W. B. Wu and X. Shao, Limit theorems for iterated random functions, J. Appl. Probab. 41 (2004), pp. 425-436.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).