The Empirical Process for Bivariate Sequences with Long Memory

We establish a functional central limit theorem for the empirical process of bivariate stationary long range dependent sequences under Gaussian subordination conditions. The proof is based upon a convergence result for cross-products of Hermite polynomials and a multivariate uniform reduction principle, as in Dehling and Taqqu [Ann. Statist. 17 (1989), 1767–1783] for the univariate case. The effect of estimated parameters is also discussed.

[1]  P. Robinson,et al.  Large-Sample Inference for Nonparametric Regression with Dependent Errors - (Now published in 'Annals of Statistics', 28 (1997), pp.2054-2083.) , 1997 .

[2]  D. Surgailis.  Long-range dependence and Appell rank , 2000 .

[3]  M. Taqqu,et al.  Whittle estimator for finite-variance non-Gaussian time series with long memory , 1999 .

[4]  Hira L. Koul,et al.  Nonparametric model checks for time series , 1999 .

[5]  P. Robinson,et al.  Time series regression with long-range dependence , 1997 .

[6]  P. Robinson.  Log-Periodogram Regression of Time Series with Long Range Dependence , 1995 .

[7]  P. Robinson,et al.  The memory of stochastic volatility models , 2001 .

[8]  Winfried Stute,et al.  Nonparametric model checks for regression , 1997 .

[9]  R. Leipus,et al.  Modelling Long‐memory Time Series with Finite or Infinite Variance: a General Approach , 2000 .

[10]  Murad S. Taqqu,et al.  Central limit theorems for quadratic forms with time-domain conditions , 1998 .

[11]  P. Massart,et al.  Invariance principles for absolutely regular empirical processes , 1995 .

[12]  M. Taqqu.  Weak convergence to fractional brownian motion and to the rosenblatt process , 1975, Advances in Applied Probability.

[13]  Domenico Marinucci,et al.  Weak convergence of multivariate fractional processes , 2000 .

[14]  Central limit theorem for the empirical process of a linear sequence with long memory , 1999 .

[15]  M. Taqqu,et al.  Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series , 1986 .

[16]  Domenico Marinucci,et al.  Alternative forms of fractional Brownian motion , 1998 .

[17]  Walter Philipp,et al.  Almost sure approximation theorems for the multivariate empirical process , 1980 .

[18]  R. Dahlhaus.  Efficient parameter estimation for self-similar processes , 1989, math/0607078.

[19]  P. Major.  Limit theorems for non-linear functionals of Gaussian sequences , 1981 .

[20]  R. Dahlhaus.  Efficient Location and Regression Estimation for Long Range Dependent Regression Models , 1995 .

[21]  D. Surgailis,et al.  A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate , 1990 .

[22]  James Durbin,et al.  Components of Cramer-von Mises statistics. I , 1972 .

[23]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[24]  M. A. Arcones,et al.  Central limit theorems for empirical andU-processes of stationary mixing sequences , 1994 .

[25]  N. Leonenko,et al.  On the Kaplan–Meier Estimator of Long-Range Dependent Sequences , 2001 .

[26]  James Durbin,et al.  Weak convergence of the sample distribution function when parameters are estimated , 1973 .

[27]  R. Dudley,et al.  Uniform Central Limit Theorems: Notation Index , 2014 .

[28]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[29]  M. A. Arcones,et al.  Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors , 1994 .

[30]  R. Dobrushin,et al.  Non-central limit theorems for non-linear functional of Gaussian fields , 1979 .

[31]  Tailen Hsing,et al.  ON THE ASYMPTOTIC EXPANSION OF THE EMPIRICAL PROCESS OF LONG-MEMORY MOVING AVERAGES , 1996 .

[32]  Winfried Stute,et al.  Bootstrap Approximations in Model Checks for Regression , 1998 .

[33]  Lixing Zhu,et al.  Model checks for regression: an innovation process approach , 1998 .

[34]  P. Robinson.  Gaussian Semiparametric Estimation of Long Range Dependence , 1995 .

[35]  M. Taqqu,et al.  The Empirical Process of some Long-Range Dependent Sequences with an Application to $U$-Statistics , 1989 .

[36]  Nikolai N. Leonenko,et al.  Statistical Analysis of Random Fields , 1989 .

[37]  Q. Shao,et al.  Weak convergence for weighted empirical processes of dependent sequences , 1996 .

[38]  M. Taqqu.  Convergence of integrated processes of arbitrary Hermite rank , 1979 .

[39]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.