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M-Dependence Approximation for Dependent Random Variables

  • Zheng-Yan LinEmail author
  • Weidong Liu
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, 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.

References

  1. 1.
    An HZ, Chen ZG, Hannan EJ (1983) The maximum of the periodogram. J Multivar Anal 13:383–400CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Anderson TW (1971) The statistical analysis of time series. Wiley, New YorkzbMATHGoogle Scholar
  3. 3.
    Andrews D (1984) Nonstrong mixing autoregressive processes. J Appl Probab 21:930–934CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Aue A (2004) Strong approximation for RCA(1) time series with applications. Stat Probab Lett 68:369–382CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Aue A, Berkes I, Horváth L (2006) Strong approximation for sums of squares of augmented GARCH sequences. Bernoulli 12:583–608CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Balan RM (2005) A strong invariance principle for associated random fields. Ann Probab 33:823–840CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Basrak B, Davis RA, Mikosch T (2002) Regular variation of GARCH processes. Stoch Proc Appl 99:95–115CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Berger E (1990) An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables. Probab Theor Relat Fields 84:161–201CrossRefzbMATHGoogle Scholar
  9. 9.
    Berkes I, Horváth L (2001) Strong approximation of the empirical process of GARCH sequences. Ann Appl Probab 11:789–809CrossRefzbMATHMathSciNetGoogle Scholar
  10. 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 41:115–137CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Berkes I, Philipp W (1979) Approximation thorems for independent and weakly dependent random vectors. Ann Probab 7:29–54CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Berkes I, Morrow GJ (1981) Strong invariance principles for mixing random fields. Probab Theor Relat Fields 57:15–37zbMATHMathSciNetGoogle Scholar
  13. 13.
    Bickel PJ, Rosenblatt M (1973) On some global measures of the deviations of density function estimates. Ann Stat 1:1071–1095CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econ 31:307–327CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Bosq D (1996) Nonparametric statistics for stochastic processes. Estimation and prediction. vol 110. Springer, New YorkGoogle Scholar
  16. 16.
    Bradley RC (1983) Approximation theorems for strongly mixing random variables. Michigan Math J 30:69–81CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Bradley RC (2005) Basic properties of strong mixing conditions. A survey and some open questions. Probab Surv 2:107–144CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Brillinger DR (1969) Asymptotic properties of spectral estimates of second order. Biometrika 56:375–390CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Chanda KC (1974) Strong mixing properties of linear stochastic processes. J Appl Probab 11:401–408CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Chanda KC (2005) Large sample properties of spectral estimators for a class of stationary nonlinear processes. J Time Ser Anal 26:1–16CrossRefzbMATHMathSciNetGoogle Scholar
  21. 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 59–71Google Scholar
  22. 22.
    Csörg M, Révész P (1981) Strong approximation in probability and statistics. Academic Press, New YorkGoogle Scholar
  23. 23.
    Csörg? M, Yu H (1996) Weak approximations for quantile processes of stationary sequences. Can J Stat 24:403–430CrossRefGoogle Scholar
  24. 24.
    Davis RA, Mikosch T (1999) The maximum of the Periodogram of a non-Gaussian sequence. Ann Probab 27:522–536CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Duan JC (1997) Augmented GARCH (p,q) process and its diffusion limit. J Econ 79:97–127CrossRefzbMATHGoogle Scholar
  26. 26.
    Eberlein E (1986) On strong invariance principles under dependence assumptions. Ann Probab 14:260–270CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Fan J, Yao Q (2003) Nonlinear time series. Nonparametric and parametric methods. Springer, New YorkCrossRefzbMATHGoogle Scholar
  28. 28.
    Fay G, Soulier P (2001) The periodogram of an i.i.d. sequence. Stoch Proc Appl 92:315–343CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Gaenssler P, Stute W (1979) Empirical processes: a survey of results for independent and identically distributed random variables. Ann Probab 7:193–243CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Györfi L, Härdle W, Sarda P, Vieu P (1989) Nonparametric curve estimation from time series. Springer, BerlinCrossRefzbMATHGoogle Scholar
  31. 31.
    Haggan V, Ozaki T (1981) Modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model. Biometrika 68:189–196CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Hsing T, Wu WB (2004) On weighted U-statistics for stationary processes. Ann Probab 32:1600–1631CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Kiefer J (1972) Skorohod embedding of multivariate RV’s and the sample DF. Probab Theor Relat Fields 24:1–35zbMATHGoogle Scholar
  34. 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 32:111–131CrossRefzbMATHGoogle Scholar
  35. 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 34:33–58CrossRefzbMATHGoogle Scholar
  36. 36.
    Kuelbs J, Philipp W (1980) Almost sure invariance principles for partial sums of mixing B-valued random variables. Ann Probab 8:1003–1036CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Lin ZY, Liu WD (2009) On maxima of periodograms of stationary processes. Ann Stat 37:2676–2695CrossRefzbMATHGoogle Scholar
  38. 38.
    Lin ZY, Lu CR (1996) Limit theory for mixing dependent random variables. Science Press, BeijingzbMATHGoogle Scholar
  39. 39.
    Liu WD (2008) Gaussian approximations for weighted empirical processes for dependent random variables. ManuscriptGoogle Scholar
  40. 40.
    Liu WD, Lin ZY (2009) Strong approximation for a class of stationary processes. Stoch Proc Appl 119:249–280CrossRefzbMATHGoogle Scholar
  41. 41.
    Liu WD, Shao QM (2009) Cramér type moderate deviation for the maximum of the periodogram with application to simultaneous tests. Ann Statist 35:1456–1486Google Scholar
  42. 42.
    Liu WD, Wu WB (2009a) Simultaneous nonparametric inference of time series. Ann StatistGoogle Scholar
  43. 43.
    Liu WD, Wu WB (2009b) Asymptotics of spectral density estimates. Econ TheorGoogle Scholar
  44. 44.
    Massart P (1989) Hungarian constructions from the nonasymptotic viewpoint. Ann Probab 17:239–256CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Mehra KL, Rao MS (1975) Weak convergence of generalized empirical processes relative to \(d_q\) under strong mixing. Ann Probab 3:979–991CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Mikosch T, Resnick S, Samorodnitsky G (2000) The maximum of the periodogram for a heavy-tailed sequence. Ann Probab 28:885–908CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Nelson DB (1990) Stationary and persistence in the GARCH(1,l) model. Econ Theor 6:318–334CrossRefGoogle Scholar
  48. 48.
    Neumann MH (1998) Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. Ann Stat 26:2014–2048CrossRefzbMATHGoogle Scholar
  49. 49.
    Philip W, Pinzur L (1980) Almost sure approximation theorems for the multivariate empirical process. Probab Theor Relat Fields 54:1–13Google Scholar
  50. 50.
    Révész P (1976) Strong approximation of the multidimensional empirical process. Ann Probab 4:729–743CrossRefzbMATHGoogle Scholar
  51. 51.
    Rio E (1995) The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann Probab 23:1188–1203CrossRefzbMATHMathSciNetGoogle Scholar
  52. 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 343–368Google Scholar
  53. 53.
    Robinson PM (1983) Nonparametric estimators for time series. J Time Ser Anal 4:185–207CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Rosenblatt M (1984) Asymptotic normality, strong mixing, and spectral density estimates. Ann Probab 12:1167–1180CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Shao QM (1993) Almost sure invariance principles for mixing sequence of random variables. Stoch Proc Appl 48:319–334CrossRefzbMATHGoogle Scholar
  56. 56.
    Shao QM, Yu H (1996) Weak convergence for weighted empirical processes of dependent sequences. Ann Probab 24:2098–2127CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    Shao X, Wu WB (2007) Asymptotic spectral theory for nonlinear time series. Ann Stat 35:1773–1801CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Strassen V (1964) An invariance principle for the law of the iterated logarithm. Z Wahrsch und Verw Gebiete 3:211–226CrossRefzbMATHMathSciNetGoogle Scholar
  59. 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 315–343Google Scholar
  60. 60.
    Tjøstheim D (1994) Non-linear time series: a selective review. Scand J Stat 21:97–130Google Scholar
  61. 61.
    Tong H (1990) Non-linear time series: a dynamical system approach. Oxford University Press, OxfordzbMATHGoogle Scholar
  62. 62.
    Wang Q, Xia YX, Gulati CM (2003) Strong approximation for long memory processes with applications. J Theor Probab 16:377–389CrossRefzbMATHGoogle Scholar
  63. 63.
    Wu WB (2005) Nonlinear system theory: another look at dependence. Proc Natl Acad Sci USA 102(40):14150–14154CrossRefzbMATHMathSciNetGoogle Scholar
  64. 64.
    Wu WB (2007) Strong invariance principles for dependent random variables. Ann Probab 35:2294–2320CrossRefzbMATHMathSciNetGoogle Scholar
  65. 65.
    Wu WB (2008) Empirical processes of stationary sequences. Stat Sinica 18:313–333zbMATHGoogle Scholar
  66. 66.
    Wu WB, Mielniczuk J (2002) Kernel density estimation for linear processes. Ann Stat 30:1441–1459CrossRefzbMATHMathSciNetGoogle Scholar
  67. 67.
    Wu WB, Shao X (2007) A limit theorem for quadratic forms and its applications. Econ Theor 23:930–951CrossRefzbMATHMathSciNetGoogle Scholar
  68. 68.
    Wu WB, Zhou Z (2011) Gaussian approximations for non-stationary multiple time series. Stat Sinica 21:1397–1413CrossRefzbMATHMathSciNetGoogle Scholar
  69. 69.
    Yu H (1996) A strong invariance principles for associated random variables. Ann Probab 24:2079–2097CrossRefzbMATHMathSciNetGoogle Scholar
  70. 70.
    Zhang LX (2004) Strong approximations of martingale vectors and their applications in Markov-chain adaptive designs. Acta Math Appl Sinica (English Series) 20:337–352CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouThe People’s Republic of China
  2. 2.Department of Mathematics and Institute of Natural SciencesShanghai Jiao Tong UniversityShanghaiThe People’s Republic of China

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