The trace problem for Toeplitz matrices and operators and its impact in probability. (English) Zbl 1348.60054

Ginovyan, Mamikon S.; Sahakyan, Artur A.; Taqqu, Murad S.

Probab. Surv. 11, 393-440 (2014).

Summary: The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by U. Grenander and G. Szegö [Toeplitz forms and their applications. Angeles: University of California Press (1958; Zbl 0080.09501)]. It has then been extensively studied in the literature. { } In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes. { } The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, hypotheses testing about the spectrum, etc. { } We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applications to discrete- and continuous-time stationary time series models with various types of memory structures, such as long memory, anti-persistent and short memory.

MSC:

60G10	Stationary stochastic processes
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
15B05	Toeplitz, Cauchy, and related matrices
60F05	Central limit and other weak theorems
60F10	Large deviations
62G05	Nonparametric estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

stationary process; Toeplitz operator; Toeplitz matrix; spectral density; central limit theorem; large deviations; estimation; trace approximation; singularity

Citations:

Zbl 0080.09501

Software:

longmemo

References:

[1]	Adenstedt, R.K. (1974). On large-sample estimation for the mean of a stationary random sequence. Ann. Math. Statist 2 1095-1107. · Zbl 0296.62081 · doi:10.1214/aos/1176342867
[2]	Anh, V.V., Angulo, J.M. and Ruiz-Medina, M.D. (1999). Possible long-range dependence in fractional random fields. J. Statist. Plann. Inference 80 95-110. · Zbl 1039.62090 · doi:10.1016/S0378-3758(98)00244-4
[3]	Anh, V.V., Leonenko, N.N. and McVinish, R. (2001). Models for fractional Riesz-Bessel motion and related processes. Fractals 9 329-346.
[4]	Avram, F. (1988). On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Related Fields 79 37-45. · Zbl 0648.60043 · doi:10.1007/BF00319101
[5]	Bentkus, R. (1972). On the error of the estimate of the spectral function of a stationary process. Lit. Mat. Sb. 12 55-71.
[6]	Bercu, B., Gamboa, F. and Rouault, A. (1997). Large deviations for quadratic forms of Gaussian stationary processes. Stochastic Process. Appl. 71 75-90. · Zbl 0941.60050 · doi:10.1016/S0304-4149(97)00071-9
[7]	Beran, J. (1994). Statistics for Long-Memory Processes . Chapman & Hall, New York. · Zbl 0869.60045
[8]	Beran, J., Feng, Y., Ghosh, S. and Kulik, R. (2013). Long-Memory Processes: Probabilistic Properties and Statistical Methods . Springer-Verlag, Berlin. · Zbl 1282.62187 · doi:10.1007/978-3-642-35512-7
[9]	Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1989). Regular Variation . Cambridge University Press, New York. · Zbl 0667.26003
[10]	Bondon, P. and Palma, W. (2006). A class of antipersistent processes. J. Time Ser. Anal. 28 261-273. · Zbl 1150.62040 · doi:10.1111/j.1467-9892.2006.00509.x
[11]	Böttcher, A. and Silbermann, B. (2005). Analysis of Toeplitz Operators , 2nd ed. Springer-Verlag, Berlin. · Zbl 0732.47029
[12]	Brockwell, P.J and Davis, R.A. (1991). Time Series: Theory and Methods , 2nd ed. Springer-Verlag, New York. · Zbl 0709.62080
[13]	Bryc, W. and Dembo, A. (1997). Large deviations for quadratic functionals of Gaussian processes. J. Theoret. Probab. 10 307-332. · Zbl 0894.60026 · doi:10.1023/A:1022656331883
[14]	Butzer, P.L. and Nessel, R.J. (1971). Fourier Analysis and Approximation I . Academic Press, New York. · Zbl 0217.42603
[15]	Coursol, J. and Dacunha-Castelle, D. (1979). Sur la formule de Chernoff pour deux processus gaussiens stationnaires. C. R. Acad. Sci. Paris Ser. A-B 288 A769-A770. · Zbl 0397.62056
[16]	Cramér, H. and Leadbetter, M.R. (1967). Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications . John Wiley & Sons, New York.
[17]	Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749-1766. · Zbl 0703.62091 · doi:10.1214/aos/1176347393
[18]	Dwight, H.B. (1961). Tables of Integrals and Other Mathematical Data . The Macmillan Company, New York.
[19]	Flandrin, P. (1989). On the spectrum of fractional Brownian motions. IEEE Trans. Inform. Theory 35 197-199. · doi:10.1109/18.42195
[20]	Fox, R. and Taqqu, M.S. (1987). Central limit theorem for quadratic forms in random variables having long-range dependence. Probab. Theory Related Fields 74 213-240. · Zbl 0586.60019 · doi:10.1007/BF00569990
[21]	Fox, R. and Taqqu, M.S. (1986). Large-sample properties of parameter estimation for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517-532. · Zbl 0606.62096 · doi:10.1214/aos/1176349936
[22]	Gamboa, F., Rouault, A. and Zani, M. (1999). A functional large deviations principle for quadratic forms of Gaussian stationary processes. Statist. Probab. Lett. 43 299-308. · Zbl 0942.60021 · doi:10.1016/S0167-7152(98)00270-3
[23]	Gao, J., Anh, V., Heyde, C. and Tieng, Q. (2001). Parameter estimation of stochastic processes with long-range dependence and intermittency. J. Time Ser. Anal. 22 517-535. · Zbl 0979.62071 · doi:10.1111/1467-9892.00239
[24]	Ginovyan, M.S. (1988). Asymptotically efficient nonparametric estimation of functionals on spectral density with zeros, Theory Probab. Appl. 33 315-322.
[25]	Ginovyan, M.S. (1988). On estimate of the value of the linear functional in a spectral density of stationary Gaussian process. Theory Probab. Appl. 33 777-781.
[26]	Ginovyan, M.S. (1993). A note on central limit theorem for Toeplitz type quadratic forms in stationary Gaussian variables. J. Cont. Math. Anal. 28 78-81. · Zbl 0846.60019
[27]	Ginovyan, M.S. (1994). On Toeplitz type quadratic functionals in Gaussian stationary process. Probab. Theory Related Fields 100 395-406. · Zbl 0817.60018 · doi:10.1007/BF01193706
[28]	Ginovyan, M.S. (1995). Asymptotic properties of spectrum estimate of stationary Gaussian process. J. Cont. Math. Anal. 30 1-17.
[29]	Ginovyan, M.S. (2003). Asymptotically efficient nonparametric estimation of nonlinear spectral functionals. Acta Appl. Math. 78 145-154. · Zbl 1029.62033 · doi:10.1023/A:1025708727313
[30]	Ginovyan, M.S. (2011). Efficient estimation of spectral functionals for Gaussian stationary models. Communications on Stochastic Analysis 5 211-232. · Zbl 1331.62193
[31]	Ginovyan, M.S. (2011). Efficient estimation of spectral functionals for continuous-time stationary models. Acta Appl. Math. 115 233-254. · Zbl 1327.62209 · doi:10.1007/s10440-011-9617-7
[32]	Ginovyan, M.S. and Sahakyan, A.A. (2005). On the central limit theorem for Toeplitz quadratic forms of stationary sequences. Theory Probab. Appl. 49 612-628. · Zbl 1090.60021 · doi:10.1137/S0040585X97981299
[33]	Ginovyan, M.S. and Sahakyan, A.A. (2007). Limit theorems for Toeplitz quadratic functionals of continuous-time stationary process. Probab. Theory Related Fields 138 551-579. · Zbl 1113.60027 · doi:10.1007/s00440-006-0037-y
[34]	Ginovyan, M.S. and Sahakyan, A.A. (2012). Trace approximations of products of truncated Toeplitz operators. Theory Probab. Appl. 56 57-71. · Zbl 1242.60022 · doi:10.1137/S0040585X97985236
[35]	Ginovyan, M.S. and Sahakyan, A.A. (2013). On the trace approximations of products of Toeplitz matrices. Statist. Probab. Lett. 83 753-760. · Zbl 1355.60044 · doi:10.1016/j.spl.2012.11.019
[36]	Giraitis, L., Koul, H. and Surgailis, D. (2012). Large Sample Inference for Long Memory Processes . Imperial College Press, London. · Zbl 1279.62016
[37]	Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate. Probab. Theory Related Fields 86 87-104. · Zbl 0717.62015 · doi:10.1007/BF01207515
[38]	Gohberg, I.C. and Krein, M.G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space . American Mathematical Society, Providence, Rhode Island. · Zbl 0181.13503
[39]	Granger, C.W.J. and Joyeux, K. (1980). An introduction to long-memory time series and fractional differencing. J. Time Ser. Anal. 1 15-29. · Zbl 0503.62079 · doi:10.1111/j.1467-9892.1980.tb00297.x
[40]	Grenander, U. and Szegö, G. (1958). Toeplitz Forms and Their Applications . University of California Press, Berkeley and Los Angeles. · Zbl 0080.09501
[41]	Guégan, D. (2005). How can we define the concept of long memory? An econometric survey. Econometric Reviews 24 113-149. · Zbl 1115.62346 · doi:10.1081/ETC-200067887
[42]	Hasminskii, R.Z. and Ibragimov, I.A. (1986). Asymptotically efficient nonparametric estimation of functionals of a spectral density function. Probab. Theory Related Fields 73 447-461. · Zbl 0595.62037 · doi:10.1007/BF00776242
[43]	Henry, M. and Zaffaroni, P. (2003). The long range dependence paradigm for macroeconomics and finance. In Long Range Dependence: Theory and Applications , Doukhan, P., Oppenheim, G., Taqqu, M.S., Eds., Birkhäuser, Boston, 417-438. · Zbl 1026.62114
[44]	Hosking, J.R.M. (2002). Fractional differencing. Biometrika. 68 165-176. · Zbl 0464.62088 · doi:10.1093/biomet/68.1.165
[45]	Ibragimov, I.A. (1963). On estimation of the spectral function of a stationary Gaussian process. Theory Probab. Appl. 8 366-401. · Zbl 0137.12901 · doi:10.1137/1108044
[46]	Ibragimov, I.A. (1967). On maximum likelihood estimation of parameters of the spectral of stationary time series. Theory Probab. Appl. 12 115-119.
[47]	Ihara, S. (2000). Large deviation theorems for Gaussian processes and their applications in information theory. Acta Appl. Math. 63 165-174. · Zbl 0983.60012 · doi:10.1023/A:1010759616411
[48]	Inoue, A. and Kasahara, Y. (1999). On the asymptotic behavior of the prediction error of a stationary process. In Trends in Probability and Related Analysis , Kono, N., Shieh, N.R., Eds., World Scientific, River Edghe, NJ, 207-218. · Zbl 0977.62100
[49]	Kac, M. (1954). Toeplitz matrices, translation kernels and a related problem in probability theory. Duke Math. J. 21 501-509. · Zbl 0056.10201 · doi:10.1215/S0012-7094-54-02149-3
[50]	Lieberman, O., Rousseau, J. and Zucker, D.M. (2001). Valid Edgeworth expansion for the sample autocorrelation function under long dependence. Econometric Theory. 17 257-275. · Zbl 1179.62029 · doi:10.1017/S0266466601171094
[51]	Lieberman, O. and Phillips, P.C.B. (2004). Error bounds and asymptotic expansions for Toeplitz product functionals of unbounded spectra. J. Time Ser. Anal. 25 733-753. · Zbl 1062.62195 · doi:10.1111/j.1467-9892.2004.01904.x
[52]	Nourdin, I. and Peccati, G. (2009). Stein’s method and exact Berry-Esséen asymptotics for functionals of Gaussian fields. Ann. Probab. 37 2231-2261. · Zbl 1196.60034 · doi:10.1214/09-AOP461
[53]	Palma, W. (2007). Long-Memory Time Series. Wiley, New York. · Zbl 1183.62153 · doi:10.1002/9780470131466
[54]	Robinson, P.M. (1995). Gaussian semiparametric estimation of long-range dependence. Ann. Statist. 23 1630-1661. · Zbl 0843.62092 · doi:10.1214/aos/1176324317
[55]	Robinson, P.M. (2003). Long memory time series. In Time Series with Long Memory , Robinson, P.M., Ed., Oxford University Press, Oxford, 4-32.
[56]	Rosenblatt, M. (1961). Independence and dependence. In Proc. 4th Berkeley Symp. Math. Stat. Probab. 2 431-443. · Zbl 0105.11802
[57]	Rosenblatt, M. (1962). Asymptotic behavior of eigenvalues of Toeplitz forms. J. Math. and Mech. 11 941-950. · Zbl 0108.31205
[58]	Rosenblatt, M. (1979). Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Z. Wahr. verw. Gebiete 49 125-132. · Zbl 0388.60048 · doi:10.1007/BF00534252
[59]	Samorodnisky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Peocesses. Chapman & Hall/CRC, New York.
[60]	Sato, T., Kakizawa, Y. and Tahiguchi, M. (1998). Large deviations results for statistics of short- and long-memory Gaussian processes. Austral. & New Zeland J. Statist. 40 17-29. · Zbl 0931.60017 · doi:10.1111/1467-842X.00003
[61]	Solo, V. (1992). Intrinsic random functions and the paradox of 1/f noise. SIAM J. Appl. Math. 52 270-291. · Zbl 0748.60043 · doi:10.1137/0152014
[62]	Taniguchi, M. (1983). On the second order asymptotic efficiency of estimators of Gaussian ARMA processes. Ann. Statist. 11 157-169. · Zbl 0509.62086 · doi:10.1214/aos/1176346066
[63]	Taniguchi, M. (1986). Berry-Esséen theorems for quadratic forms of Gaussian stationary processes. Probab. Theory Related Fields 72 185-194. · Zbl 0572.60030 · doi:10.1007/BF00699102
[64]	Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series. Academic Press, New York. · Zbl 0955.62088
[65]	Taqqu, M.S. (1975). Weak convergence to fractional Brownian motion and the Rosenblatt process. Z. Wahr. verw. Gebiete 31 287-302. · Zbl 0303.60033 · doi:10.1007/BF00532868
[66]	Taqqu, M.S. (1987). Toeplitz matrices and estimation of time series with long-range dependence. In Proc. of the First World Congress of the Bernoulli Society. VNU Science Press, BV: Utrecht, The Netherlands, 1 75-83.
[67]	Taqqu, M.S. (2003). Fractional Brownian motion and long-range dependence. In Long Range Dependence: Theory and Applications , Doukhan, P., Oppenheim, G., Taqqu, M.S., Eds., Birkhäuser, Boston, 5-38. · Zbl 1039.60041
[68]	Taqqu, M.S. (2011). The Rosenblatt process. In Selected Works of Murray Rosenblatt, Selected Works in Probab. and Statist. , Davis, R.A. et al., Eds., Springer, Boston, 29-45.
[69]	Terrin, N. and Taqqu, M.S. (1990). A noncentral limit theorem for quadratic forms of Gaussian stationary sequences. J. Theoret. Probab. 3 449-475. · Zbl 0724.60048 · doi:10.1007/BF01061262
[70]	Terrin, N. and Taqqu, M.S. (1991). Convergence to a Gaussian limit as the normalization exponent tends to 1/2. Statist. Probab. Lett. 11 419-427. · Zbl 0729.60017 · doi:10.1016/0167-7152(91)90191-S
[71]	Veillette, M. and Taqqu, M.S. (2013). Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19 982-1005. · Zbl 1273.60020 · doi:10.3150/12-BEJ421
[72]	Yaglom, A.M. (1986). The Correlation Theory of Stationary and Related Random Processes, Vol. 1. Springer, New York.