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.
Keywords
- Dependent Random Variables
- Strong Invariance Principle (SIP)
- Spectral Density Estimation
- Periodogram
- Empirical Process
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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An HZ, Chen ZG, Hannan EJ (1983) The maximum of the periodogram. J Multivar Anal 13:383–400
Anderson TW (1971) The statistical analysis of time series. Wiley, New York
Andrews D (1984) Nonstrong mixing autoregressive processes. J Appl Probab 21:930–934
Aue A (2004) Strong approximation for RCA(1) time series with applications. Stat Probab Lett 68:369–382
Aue A, Berkes I, Horváth L (2006) Strong approximation for sums of squares of augmented GARCH sequences. Bernoulli 12:583–608
Balan RM (2005) A strong invariance principle for associated random fields. Ann Probab 33:823–840
Basrak B, Davis RA, Mikosch T (2002) Regular variation of GARCH processes. Stoch Proc Appl 99:95–115
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
Berkes I, Horváth L (2001) Strong approximation of the empirical process of GARCH sequences. Ann Appl Probab 11:789–809
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
Berkes I, Philipp W (1979) Approximation thorems for independent and weakly dependent random vectors. Ann Probab 7:29–54
Berkes I, Morrow GJ (1981) Strong invariance principles for mixing random fields. Probab Theor Relat Fields 57:15–37
Bickel PJ, Rosenblatt M (1973) On some global measures of the deviations of density function estimates. Ann Stat 1:1071–1095
Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econ 31:307–327
Bosq D (1996) Nonparametric statistics for stochastic processes. Estimation and prediction. vol 110. Springer, New York
Bradley RC (1983) Approximation theorems for strongly mixing random variables. Michigan Math J 30:69–81
Bradley RC (2005) Basic properties of strong mixing conditions. A survey and some open questions. Probab Surv 2:107–144
Brillinger DR (1969) Asymptotic properties of spectral estimates of second order. Biometrika 56:375–390
Chanda KC (1974) Strong mixing properties of linear stochastic processes. J Appl Probab 11:401–408
Chanda KC (2005) Large sample properties of spectral estimators for a class of stationary nonlinear processes. J Time Ser Anal 26:1–16
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
Csörg M, Révész P (1981) Strong approximation in probability and statistics. Academic Press, New York
Csörg? M, Yu H (1996) Weak approximations for quantile processes of stationary sequences. Can J Stat 24:403–430
Davis RA, Mikosch T (1999) The maximum of the Periodogram of a non-Gaussian sequence. Ann Probab 27:522–536
Duan JC (1997) Augmented GARCH (p,q) process and its diffusion limit. J Econ 79:97–127
Eberlein E (1986) On strong invariance principles under dependence assumptions. Ann Probab 14:260–270
Fan J, Yao Q (2003) Nonlinear time series. Nonparametric and parametric methods. Springer, New York
Fay G, Soulier P (2001) The periodogram of an i.i.d. sequence. Stoch Proc Appl 92:315–343
Gaenssler P, Stute W (1979) Empirical processes: a survey of results for independent and identically distributed random variables. Ann Probab 7:193–243
Györfi L, Härdle W, Sarda P, Vieu P (1989) Nonparametric curve estimation from time series. Springer, Berlin
Haggan V, Ozaki T (1981) Modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model. Biometrika 68:189–196
Hsing T, Wu WB (2004) On weighted U-statistics for stationary processes. Ann Probab 32:1600–1631
Kiefer J (1972) Skorohod embedding of multivariate RV’s and the sample DF. Probab Theor Relat Fields 24:1–35
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
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
Kuelbs J, Philipp W (1980) Almost sure invariance principles for partial sums of mixing B-valued random variables. Ann Probab 8:1003–1036
Lin ZY, Liu WD (2009) On maxima of periodograms of stationary processes. Ann Stat 37:2676–2695
Lin ZY, Lu CR (1996) Limit theory for mixing dependent random variables. Science Press, Beijing
Liu WD (2008) Gaussian approximations for weighted empirical processes for dependent random variables. Manuscript
Liu WD, Lin ZY (2009) Strong approximation for a class of stationary processes. Stoch Proc Appl 119:249–280
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
Liu WD, Wu WB (2009a) Simultaneous nonparametric inference of time series. Ann Statist
Liu WD, Wu WB (2009b) Asymptotics of spectral density estimates. Econ Theor
Massart P (1989) Hungarian constructions from the nonasymptotic viewpoint. Ann Probab 17:239–256
Mehra KL, Rao MS (1975) Weak convergence of generalized empirical processes relative to \(d_q\) under strong mixing. Ann Probab 3:979–991
Mikosch T, Resnick S, Samorodnitsky G (2000) The maximum of the periodogram for a heavy-tailed sequence. Ann Probab 28:885–908
Nelson DB (1990) Stationary and persistence in the GARCH(1,l) model. Econ Theor 6:318–334
Neumann MH (1998) Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. Ann Stat 26:2014–2048
Philip W, Pinzur L (1980) Almost sure approximation theorems for the multivariate empirical process. Probab Theor Relat Fields 54:1–13
Révész P (1976) Strong approximation of the multidimensional empirical process. Ann Probab 4:729–743
Rio E (1995) The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann Probab 23:1188–1203
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
Robinson PM (1983) Nonparametric estimators for time series. J Time Ser Anal 4:185–207
Rosenblatt M (1984) Asymptotic normality, strong mixing, and spectral density estimates. Ann Probab 12:1167–1180
Shao QM (1993) Almost sure invariance principles for mixing sequence of random variables. Stoch Proc Appl 48:319–334
Shao QM, Yu H (1996) Weak convergence for weighted empirical processes of dependent sequences. Ann Probab 24:2098–2127
Shao X, Wu WB (2007) Asymptotic spectral theory for nonlinear time series. Ann Stat 35:1773–1801
Strassen V (1964) An invariance principle for the law of the iterated logarithm. Z Wahrsch und Verw Gebiete 3:211–226
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
Tjøstheim D (1994) Non-linear time series: a selective review. Scand J Stat 21:97–130
Tong H (1990) Non-linear time series: a dynamical system approach. Oxford University Press, Oxford
Wang Q, Xia YX, Gulati CM (2003) Strong approximation for long memory processes with applications. J Theor Probab 16:377–389
Wu WB (2005) Nonlinear system theory: another look at dependence. Proc Natl Acad Sci USA 102(40):14150–14154
Wu WB (2007) Strong invariance principles for dependent random variables. Ann Probab 35:2294–2320
Wu WB (2008) Empirical processes of stationary sequences. Stat Sinica 18:313–333
Wu WB, Mielniczuk J (2002) Kernel density estimation for linear processes. Ann Stat 30:1441–1459
Wu WB, Shao X (2007) A limit theorem for quadratic forms and its applications. Econ Theor 23:930–951
Wu WB, Zhou Z (2011) Gaussian approximations for non-stationary multiple time series. Stat Sinica 21:1397–1413
Yu H (1996) A strong invariance principles for associated random variables. Ann Probab 24:2079–2097
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
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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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DOI: https://doi.org/10.1007/978-1-4614-1966-2_9
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