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M-Dependence Approximation for Dependent Random Variables

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Probability Approximations and Beyond

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 205))

Abstract

The purpose of this paper is to describe the m-dependence approximation and some recent results obtained by using the m-dependence approximation technique. In particular, we will focus on strong invariance principles of the partial sums and empirical processes, kernel density estimation, spectral density estimation and the theory on periodogram. This paper is an update of, and a supplement to the paper “m-Dependent Approximation” by the authors in The International Congress of Chinese Mathematicians (ICCM) 2007, Vol II, 720–734.

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References

  1. An HZ, Chen ZG, Hannan EJ (1983) The maximum of the periodogram. J Multivar Anal 13:383–400

    Article  MATH  MathSciNet  Google Scholar 

  2. Anderson TW (1971) The statistical analysis of time series. Wiley, New York

    MATH  Google Scholar 

  3. Andrews D (1984) Nonstrong mixing autoregressive processes. J Appl Probab 21:930–934

    Article  MATH  MathSciNet  Google Scholar 

  4. Aue A (2004) Strong approximation for RCA(1) time series with applications. Stat Probab Lett 68:369–382

    Article  MATH  MathSciNet  Google Scholar 

  5. Aue A, Berkes I, Horváth L (2006) Strong approximation for sums of squares of augmented GARCH sequences. Bernoulli 12:583–608

    Article  MATH  MathSciNet  Google Scholar 

  6. Balan RM (2005) A strong invariance principle for associated random fields. Ann Probab 33:823–840

    Article  MATH  MathSciNet  Google Scholar 

  7. Basrak B, Davis RA, Mikosch T (2002) Regular variation of GARCH processes. Stoch Proc Appl 99:95–115

    Article  MATH  MathSciNet  Google Scholar 

  8. Berger E (1990) An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables. Probab Theor Relat Fields 84:161–201

    Article  MATH  Google Scholar 

  9. Berkes I, Horváth L (2001) Strong approximation of the empirical process of GARCH sequences. Ann Appl Probab 11:789–809

    Article  MATH  MathSciNet  Google Scholar 

  10. Berkes I, Philipp W (1977) An almost sure invariance principle for the empirical distribution function of mixing random variables. Z Wahrsch und Verw Gebiete 41:115–137

    Article  MATH  MathSciNet  Google Scholar 

  11. Berkes I, Philipp W (1979) Approximation thorems for independent and weakly dependent random vectors. Ann Probab 7:29–54

    Article  MATH  MathSciNet  Google Scholar 

  12. Berkes I, Morrow GJ (1981) Strong invariance principles for mixing random fields. Probab Theor Relat Fields 57:15–37

    MATH  MathSciNet  Google Scholar 

  13. Bickel PJ, Rosenblatt M (1973) On some global measures of the deviations of density function estimates. Ann Stat 1:1071–1095

    Article  MATH  MathSciNet  Google Scholar 

  14. Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econ 31:307–327

    Article  MATH  MathSciNet  Google Scholar 

  15. Bosq D (1996) Nonparametric statistics for stochastic processes. Estimation and prediction. vol 110. Springer, New York

    Google Scholar 

  16. Bradley RC (1983) Approximation theorems for strongly mixing random variables. Michigan Math J 30:69–81

    Article  MATH  MathSciNet  Google Scholar 

  17. Bradley RC (2005) Basic properties of strong mixing conditions. A survey and some open questions. Probab Surv 2:107–144

    Article  MATH  MathSciNet  Google Scholar 

  18. Brillinger DR (1969) Asymptotic properties of spectral estimates of second order. Biometrika 56:375–390

    Article  MATH  MathSciNet  Google Scholar 

  19. Chanda KC (1974) Strong mixing properties of linear stochastic processes. J Appl Probab 11:401–408

    Article  MATH  MathSciNet  Google Scholar 

  20. Chanda KC (2005) Large sample properties of spectral estimators for a class of stationary nonlinear processes. J Time Ser Anal 26:1–16

    Article  MATH  MathSciNet  Google Scholar 

  21. Csörg P, Révész P (1975) Some notes on the empirical distribution function and the quantile process. In: Revesz P (ed) Limit theorems of probability theory, vol 11. North-Holland, Amsterdam, pp 59–71

    Google Scholar 

  22. Csörg M, Révész P (1981) Strong approximation in probability and statistics. Academic Press, New York

    Google Scholar 

  23. Csörg? M, Yu H (1996) Weak approximations for quantile processes of stationary sequences. Can J Stat 24:403–430

    Article  Google Scholar 

  24. Davis RA, Mikosch T (1999) The maximum of the Periodogram of a non-Gaussian sequence. Ann Probab 27:522–536

    Article  MATH  MathSciNet  Google Scholar 

  25. Duan JC (1997) Augmented GARCH (p,q) process and its diffusion limit. J Econ 79:97–127

    Article  MATH  Google Scholar 

  26. Eberlein E (1986) On strong invariance principles under dependence assumptions. Ann Probab 14:260–270

    Article  MATH  MathSciNet  Google Scholar 

  27. Fan J, Yao Q (2003) Nonlinear time series. Nonparametric and parametric methods. Springer, New York

    Book  MATH  Google Scholar 

  28. Fay G, Soulier P (2001) The periodogram of an i.i.d. sequence. Stoch Proc Appl 92:315–343

    Article  MATH  MathSciNet  Google Scholar 

  29. Gaenssler P, Stute W (1979) Empirical processes: a survey of results for independent and identically distributed random variables. Ann Probab 7:193–243

    Article  MATH  MathSciNet  Google Scholar 

  30. Györfi L, Härdle W, Sarda P, Vieu P (1989) Nonparametric curve estimation from time series. Springer, Berlin

    Book  MATH  Google Scholar 

  31. Haggan V, Ozaki T (1981) Modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model. Biometrika 68:189–196

    Article  MATH  MathSciNet  Google Scholar 

  32. Hsing T, Wu WB (2004) On weighted U-statistics for stationary processes. Ann Probab 32:1600–1631

    Article  MATH  MathSciNet  Google Scholar 

  33. Kiefer J (1972) Skorohod embedding of multivariate RV’s and the sample DF. Probab Theor Relat Fields 24:1–35

    MATH  Google Scholar 

  34. Komlós J, Major P, Tusnády G (1975) An approximation of partial sums of independent RV’s and the sample DF. I. Z. Wahrsch und Verw Gebiete 32:111–131

    Article  MATH  Google Scholar 

  35. Komlós J, Major P, Tusnády G (1976) An approximation of partial sums of independent RV’s and the sample DF. II. Z. Wahrsch und Verw Gebiete 34:33–58

    Article  MATH  Google Scholar 

  36. Kuelbs J, Philipp W (1980) Almost sure invariance principles for partial sums of mixing B-valued random variables. Ann Probab 8:1003–1036

    Article  MATH  MathSciNet  Google Scholar 

  37. Lin ZY, Liu WD (2009) On maxima of periodograms of stationary processes. Ann Stat 37:2676–2695

    Article  MATH  Google Scholar 

  38. Lin ZY, Lu CR (1996) Limit theory for mixing dependent random variables. Science Press, Beijing

    MATH  Google Scholar 

  39. Liu WD (2008) Gaussian approximations for weighted empirical processes for dependent random variables. Manuscript

    Google Scholar 

  40. Liu WD, Lin ZY (2009) Strong approximation for a class of stationary processes. Stoch Proc Appl 119:249–280

    Article  MATH  Google Scholar 

  41. Liu WD, Shao QM (2009) Cramér type moderate deviation for the maximum of the periodogram with application to simultaneous tests. Ann Statist 35:1456–1486

    Google Scholar 

  42. Liu WD, Wu WB (2009a) Simultaneous nonparametric inference of time series. Ann Statist

    Google Scholar 

  43. Liu WD, Wu WB (2009b) Asymptotics of spectral density estimates. Econ Theor

    Google Scholar 

  44. Massart P (1989) Hungarian constructions from the nonasymptotic viewpoint. Ann Probab 17:239–256

    Article  MATH  MathSciNet  Google Scholar 

  45. Mehra KL, Rao MS (1975) Weak convergence of generalized empirical processes relative to \(d_q\) under strong mixing. Ann Probab 3:979–991

    Article  MATH  MathSciNet  Google Scholar 

  46. Mikosch T, Resnick S, Samorodnitsky G (2000) The maximum of the periodogram for a heavy-tailed sequence. Ann Probab 28:885–908

    Article  MATH  MathSciNet  Google Scholar 

  47. Nelson DB (1990) Stationary and persistence in the GARCH(1,l) model. Econ Theor 6:318–334

    Article  Google Scholar 

  48. Neumann MH (1998) Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. Ann Stat 26:2014–2048

    Article  MATH  Google Scholar 

  49. Philip W, Pinzur L (1980) Almost sure approximation theorems for the multivariate empirical process. Probab Theor Relat Fields 54:1–13

    Google Scholar 

  50. Révész P (1976) Strong approximation of the multidimensional empirical process. Ann Probab 4:729–743

    Article  MATH  Google Scholar 

  51. Rio E (1995) The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann Probab 23:1188–1203

    Article  MATH  MathSciNet  Google Scholar 

  52. Robinson PM (1983) Review of various approaches to power spectrum estimation. In: Brillinger DR, Krishnaiah RR (eds) Time series in the frequency domain. Handbook of statistics. vol 3. North-Holland, Amsterdam, pp 343–368

    Google Scholar 

  53. Robinson PM (1983) Nonparametric estimators for time series. J Time Ser Anal 4:185–207

    Article  MATH  MathSciNet  Google Scholar 

  54. Rosenblatt M (1984) Asymptotic normality, strong mixing, and spectral density estimates. Ann Probab 12:1167–1180

    Article  MATH  MathSciNet  Google Scholar 

  55. Shao QM (1993) Almost sure invariance principles for mixing sequence of random variables. Stoch Proc Appl 48:319–334

    Article  MATH  Google Scholar 

  56. Shao QM, Yu H (1996) Weak convergence for weighted empirical processes of dependent sequences. Ann Probab 24:2098–2127

    Article  MATH  MathSciNet  Google Scholar 

  57. Shao X, Wu WB (2007) Asymptotic spectral theory for nonlinear time series. Ann Stat 35:1773–1801

    Article  MATH  MathSciNet  Google Scholar 

  58. Strassen V (1964) An invariance principle for the law of the iterated logarithm. Z Wahrsch und Verw Gebiete 3:211–226

    Article  MATH  MathSciNet  Google Scholar 

  59. Strassen V (1967) Almost sure behaviour of sums of independent random variables and martingales. Proceedings of the 5th Berkeley symposium of mathematical statistics and probability, vol 2. University of California Press, Berkeley, pp 315–343

    Google Scholar 

  60. Tjøstheim D (1994) Non-linear time series: a selective review. Scand J Stat 21:97–130

    Google Scholar 

  61. Tong H (1990) Non-linear time series: a dynamical system approach. Oxford University Press, Oxford

    MATH  Google Scholar 

  62. Wang Q, Xia YX, Gulati CM (2003) Strong approximation for long memory processes with applications. J Theor Probab 16:377–389

    Article  MATH  Google Scholar 

  63. Wu WB (2005) Nonlinear system theory: another look at dependence. Proc Natl Acad Sci USA 102(40):14150–14154

    Article  MATH  MathSciNet  Google Scholar 

  64. Wu WB (2007) Strong invariance principles for dependent random variables. Ann Probab 35:2294–2320

    Article  MATH  MathSciNet  Google Scholar 

  65. Wu WB (2008) Empirical processes of stationary sequences. Stat Sinica 18:313–333

    MATH  Google Scholar 

  66. Wu WB, Mielniczuk J (2002) Kernel density estimation for linear processes. Ann Stat 30:1441–1459

    Article  MATH  MathSciNet  Google Scholar 

  67. Wu WB, Shao X (2007) A limit theorem for quadratic forms and its applications. Econ Theor 23:930–951

    Article  MATH  MathSciNet  Google Scholar 

  68. Wu WB, Zhou Z (2011) Gaussian approximations for non-stationary multiple time series. Stat Sinica 21:1397–1413

    Article  MATH  MathSciNet  Google Scholar 

  69. Yu H (1996) A strong invariance principles for associated random variables. Ann Probab 24:2079–2097

    Article  MATH  MathSciNet  Google Scholar 

  70. Zhang LX (2004) Strong approximations of martingale vectors and their applications in Markov-chain adaptive designs. Acta Math Appl Sinica (English Series) 20:337–352

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, The People’s Republic of China

    Zheng-Yan Lin

  2. Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, The People’s Republic of China

    Weidong Liu

Authors
  1. Zheng-Yan Lin
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  2. Weidong Liu
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Correspondence to Zheng-Yan Lin .

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Editors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, Zürich, 8057, Switzerland

    Andrew Barbour

  2. , Department of Statistics and Applied Pro, National University of Singapore, Singapore, 119260, Singapore

    Hock Peng Chan

  3. Dept. Statistics, Stanford University, Serra Mall 390, Stanford, 94305, California, USA

    David Siegmund

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Lin, ZY., Liu, W. (2012). M-Dependence Approximation for Dependent Random Variables. In: Barbour, A., Chan, H., Siegmund, D. (eds) Probability Approximations and Beyond. Lecture Notes in Statistics(), vol 205. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1966-2_9

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