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 Probability Surveys > Vol. 9 (2012) open journal systems 


Multivariate prediction and matrix Szegö theory

Nicholas H. Bingham, Imperial College, London


Abstract
Following the recent survey by the same author of Szegö's theorem and orthogonal polynomials on the unit circle (OPUC) in the scalar case, we survey the corresponding multivariate prediction theory and matrix OPUC (MOPUC).

AMS 2000 subject classifications: Primary 60G10; secondary 60G25.

Keywords: Stationary process, vector-valued, multivariate prediction theory, multivariate orthogonal polynomials on the unit circle (MOPUC), Kolmogorov isomorphism theorem, Verblunsky’s theorem, Szegö’s theorem.

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Bingham, Nicholas H., Multivariate prediction and matrix Szegö theory, Probability Surveys, 9, (2012), 325-339 (electronic). DOI: 10.1214/12-PS200.

References

[AdArKr]     Adamjan, V. M., Arov, D. Z. and Krein, M. G., Infinite Hankel matrices and generalized problems of Carathéodory–Fejér and I. Schur. Functional Anal. Appl. 2 (1968), 269-281.

[ArDy1]     Arov, D. Z. and Dym, H., Matricial Nehari problems, J-inner functions and the Muckenhoupt condition. J. Funct. Anal. 181 (2001), 227-299. MR1821698

[ArDy2]     Arov, D. Z. and Dym, H., Criteria for the strong regularity of J-inner functions and γ-generating functions. J. Math. Anal. Appl. 280 (2003), 387-399. MR1977919

[ArDy3]     Arov, D. Z. and Dym, H., J-contractive matrix valued functions and related topics, Encycl. Math. Appl. 116, Cambridge Univ. Press, Cambridge, 2008. MR2474532

[Bar1]     Barclay, S. J., Continuity of the spectral factorization maping. J. London Math. Soc. 70 (2004), 763-779. MR2096876

[Bar2]     Barclay, S. J., A solution of the Douglas-Rudin problem for matrix-valued functions. Proc. London Math. Soc. 99 (2009), 757-786. MR2551470

[BasWi]     Basor, E. L. and Widom, H., On a Toeplitz determinant identity of Borodin and Okounkov. Integral Equations and Operator Theory 37 (2000), 397-401. MR1780119

[Bax1]     Baxter, G., A convergence equivalence related to polynomials orthogonal on the unit circle. Trans. Amer. Math. Soc. 99 (1961), 471-487. MR0126126

[Bax2]     Baxter, G., An asymptotic result for the finite predictor. Math. Scand. 10 (1962), 137-144. MR0149584

[Bax3]     Baxter, G., A norm inequality for a “finite-section” Wiener-Hopf equation. Illinois J. Math. 7 (1963), 97-103. MR0145285

[Beu]     Beurling, A., On two problems concerning linear transformations in Hilbert space. Acta Math. 81 (1948), 239-255 (reprinted in The collected works of Arne Beurling, Volumes 1,2, Birkhäuser, 1989). MR0027954

[Bi]     Bingham, N. H., Szegö’s theorem and its probabilistic descendants. Probability Surveys 9 (2012), 287-324.

[BiFr]     Bingham, N. H. and Fry, J. M., Regression: Linear models in statistics. SUMS (Springer Undergraduate Mathematics Series), 2010. MR2724817

[BiFrKi]     Bingham, N. H., Fry, J. M. and Kiesel, R., Multivariate elliptic processes. Statistica Neerlandica 64 (2010), 352-366. MR2683465

[BiInKa]     Bingham, N. H., Inoue, A. and Kasahara, Y., An explicit representation of Verblunsky coefficients. Statistics and Probability Letters 82.2 (2012), 403-410. MR2875229

[BlJeHa]     Bloomfield, P., Jewell, N. P. and Hayashi, E., Characterization of completely non-deterministic stochastic processes. Pacific J. Math. 107 (1983), 307-317. MR0705750

[Bo1]     Böttcher, A., One more proof of the Borodin-Okounkiv formula for Toeplitz determinants. Integral Equations and Operator Theory 41 (2001), 13 -125. MR1844464

[Bo2]     Böttcher, A., Featured review of the Borodin-Okounkov and Basor-Widom papers. Mathematical Reviews 1790118/9 (2001g:47042a,b).

[Bo3]     Böttcher, A., On the determinant formulas by Borodin, Okounkov, Baik, Deift and Rains. Operator Th. Adv. Appl. 135, 91-99, Birkhäuser, Basel, 2002. MR1935759

[Cha1]     Chanzy, J., Théorèmes-limite de Szëgo dans le cas matriciel. Proc. Japan Acad. A 82 (2006), 113-116. MR2279275

[Cha2]     Chanzy, J., Inversion d’un opérateur de Toeplitz tronqué à symbole matriciel et théorèmes-limite de Szegö. Ann. Math. Blaise Pascal 13 (2006), 111-205. MR2233014

[ChePou]     Cheng, R. and Pourahmadi, M., Baxter’s inequality and convergence of predictors of multivariate stochastic processes. Probab. Th. Rel. Fields 95 (1993), 115-124. MR1207310

[CuZw]     Curtain, R. F. and Zwart, H., An introduction to infinite-dimensional linear systems. Springer, 1995. MR1351248

[DaPuSi]     Damanik, D., Pushnitski, A. and Simon, B., The analytic theory of matrix orthogonal polynomials. Surveys in Approximation Theory 4 (2008), 1-85. MR2379691

[DeHoKhTy]     Derevyagin, M., Holtz, O., Khrushchev, S. and Tyaglov, M., Szegö’s theorem for matrix orthogonal polynomials. arXiv:1104.4999v1 [math.CA] 26 April 2011.

[Dev]     Devinatz, A., The factorization of operator-valued functions. Ann. Math. 73 (1961), 458-495. MR0126702

[DuFrKi]     Dubovoj, V. K., Fritzsche, B. and Kirstein, B., Matricial version of the classical Schur problem. Teubner, Stuttgart, 1992.

[EpJaLa]     Ephremidze, L., Janashia, G. and Lagvilava, E., An analytic proof of the matrix spectral factorization theorem. Georgian Math. J. 15 (2008), 241-249. MR2428468

[EpLa]     Ephremidze, L. and Lagvilada, E., Remark on outer analytic matrix functions. Proc. A. Razmadze Math. Inst. 152 (2010), 29-32. MR2663529

[FrBh]     Frazho, A. E. and Bhosri, W., An operator perspective on signals and systems. Operator Theory: advances and Applications 204, Birkhäuser, 2010. MR2584037

[FoFr]     Foias, C. and Frazho, A. E., The commutant lifting approach to interpolation problems. Operator Theory: Advances and Applications 44, Birkhäuser, 1990. MR1120546

[GrSz]     Grenander, U. and Szegö, G., Toeplitz forms and their applications. Univ. California Press, Berkeley, CA, 1958. MR0094840

[Hal]     Halmos, P. R., Shifts on Hilbert spaces. J. reine ang. Math. 208 (1961), 102-112. MR0152896

[Ha]     Hannan, E. J., Multiple time series, Wiley, 1970 MR0279952

[HeLo]     Helson, H. and Lowdenslager, D., Prediction theory and Fourier series in several variables, I, II. Acta Math. 99 (1958), 165-202, 106 (1961), 175-213. MR0097688

[HeSa]     Helson, H. and Sarason, D., Past and future. Math. Scand 21 (1967), 5-16. MR0236989

[HeSz]     H. Helson and G. Szegö, A problem in prediction theory. Acta Mat. Pura Appl. 51 (1960), 107-138. MR0121608

[HiPh]     Hille, E. and Phillips, R. S., Functional anaysis and semigroups. Colloq. Publ. 31, American Math. Soc., 1957. MR0089373

[IbRo]     Ibragimov, I. A. and Rozanov, Yu. A., Gaussian random processes. Springer, 1978. MR0543837

[In1]     Inoue, A., Asymptotics for the partial autocorrelation function of a stationary process. J. Analyse Math. 81 (2000), 65-109. MR1785278

[In2]     Inoue, A., Asymptotic behaviour for partial autocorrelation functions of fractional ARIMA processes. Ann. Appl. Prob. 12 (2002), 1471-1491. MR1936600

[In3]     Inoue, A., AR and MA presentations of partial autocorrelation functions with applications. Prob. Th. Rel. Fields 140 (2008), 523-551. MR2365483

[InKa1]     Inoue, A. and Kasahara, Y., Partial autocorrelation functions of the fractional ARIMA process. J. Multivariate Analysis 89 (2004), 135-147. MR2041213

[InKa2]     Inoue, A. and Kasahara, Y., Explicit representation of finite predictor coefficients and its applications. Ann. Statist. 34 (2006), 973-993. MR2283400

[JaLaEp]     Janashia, G., Lagvilada, E. and Ephremidze, L., A new method of matrix spectral factorization. IEEE Trans. Info. Th. 57 (2011), 2318-2326. MR2809092

[Kal]     Kallenberg, O., Foundations of modern probability, 2nd ed., Springer, 2002. MR1876169

[KasBi]     Kasahara, Y. and Bingham, N. H., Verblunsky coefficients and Nehari sequences. Trans. Amer. Math. Soc., to appear.

[Kol]     Kolmogorov, A. N., Stationary sequences in Hilbert space. Bull. Moskov. Gos. Univ. Mat. 2 (1941), 1-40 (in Russian; reprinted, Selected works of A. N. Kolmogorov, Vol. 2: Theory of probability and mathematical statistics, Nauka, Moskva, 1986, 215-255).

[Lax1]     Lax, P. D., Translation-invariant subspaces. Acta Math. 101 (1959), 163-178. MR0105620

[Lax2]     Lax, P. D., On the regularity of spectral densities. Th. Probab. Appl. 8 (1963), 337-340. MR0156217

[Lax3]     Lax, P. D., On the factorization of matrix-valued functions. Comm. Pure Appl. Math. 29 (1976), 683-688. MR0425663

[LeMcK]     Levinson, N. and McKean, H. P., Weighted trigonometrical approximation on R1 with application to the germ field of a stationary Gaussian noise. Acta Math. 112 (1964), 99-143 (reprinted in Selected papers of Norman Levinson Vol. 2 (Birkhäuser, Basel, 1998), IX, 222-266). MR0163111

[MakMiSc]     Makagon, A., Miamee, A. G. and Schröder, B. S. W., Recursive condition for positivity of the angle for multivariate stationary sequences. Proc. Amer. Math. Soc. 126 (1998), 1821-1825. MR1443841

[MakSa]     Makagon, A. and Salehi, H., Notes on infinite-dimensional stationary sequences. Probability Theory on Vector Spaces IV, 200-238. Lecture Notes in Math. 1391, Springer, 1987. MR1020565

[MakWe]     Makagon, A. and Weron, A., q-variate minimal stationary processes. Studia Math. 59 (1976), 41-52. MR0428419

[Mas1]     Masani, P., Cramér’s theorem on monotone matrix functions and the Wold decomposition. Probability and statistics: The Harald Cramér volume (ed. U. Grenander) 175-189, Wiley, 1959. MR0124929

[Mas2]     Masani, P., The prediction theory of multivariate stochastic processes, III. Acta Math. 104 (1960), 141-162. MR0121952

[Mas3]     Masani, P., Shift-invariant spaces and prediction theory. Acta Math. 107 (1962), 275-290. MR0140930

[Mas4]     Masani, P., Recent trends in multivariate prediction theory. Multivariate Analysis (Proc. Int. Symp., Dayton OH) 351-382, Academic Press, 1966. MR0214228

[Mas5]     Masani, P., Comments on the prediction-theoretic papers, 276-305 in [Wi].

[Mat]     Matveev, R. F., Regularity of multi-dimensional stochastic processes with discrete time. Dokl. Akad. Nauk SSSR 126 (1959), 713-715. MR0115215

[Mei]     Tao Mei, Operator-valued Hardy spaces. Memoirs Amer. Math. Soc. vol. 188 no. 881, 2007. MR2327840

[MoViKa]     Morf, M., Vieira, A. and Kailath, T., Covariance characterization by partial autocorrelation matrices. Ann. Statist. 6 (1978), 643-678. MR0478519

[Nik1]     Nikolskii, N. K., Treatise on the shifgt operator: Spectral function theory. Grundl. math. Wiss. 237, Springer, 1986. MR0827223

[Nik2]     Nikolskii, N. K., Operators, functions and systems: an easy reading. Volume 1: Hardy, Hankel and Toeplitz; Volume 2: Model operators and systems. Math. Surveys and Monographs 92, 93, Amer. Math. Soc., 2002. MR1864396

[Pel1]     Peller, V. V., Hankel operators and multivariate stochastic processes. Proc. Symp. Pure Math. 51, Part 1, 357-371, AMS, Providence, RI, 1990. MR1077396

[Pel2]     Peller, V. V., Factorization and approximation problems for matrix functions. J. American Math. soc. 11 (1998), 751-770. MR1618768

[Pel3]     Peller, V. V., Hankel operators and their applications, Springer, 2003. MR1949210

[Pou1]     Pourahmadi, M., A matricial extension of the Helson-Szegö theorem and its application in multivariate prediction. J. Multivariate Analysis 16 (1985), 265-275. MR0790606

[Pou2]     Pourahmadi, M., Joint mean-covariance models with applications to longitudinal data. Unconstrained parametrization. Biometrika 86 (1999), 677-690. MR1723786

[Pou3]     Pourahmadi, M., Foundations of time series analysis and prediction theory. Wiley, 2001. MR1849562

[Re]     Reinsel, G. C., Elements of multivariate time series analysis, Springer, 1997. MR1451875

[RiNa]     Riesz, F. and Sz.-Nagy, B., Leçons d’analyse fonctionnelle, 2nd ed., Akad. Kiadó, 1953. MR0056821

[RosRov]     Rosenblum, M. and Rovnyak, J., Hardy classes and operator theory, Dover, New York, 1997 (1st ed. Oxford University Press, 1985). MR1435287

[Ro1]     Rozanov, Yu. A., Spectral properties of multivariate stationary processes and boundary properties of analytic functions. Th. Probab. Appl. 5 (1960), 362-376.

[Ro2]     Rozanov, Yu. A., Stationary random processes, Holden Day, San Francisco, CA, 1967. MR0214134

[Sa1]     Sarason, D., An addendum to “Past and future”. Math. Scand. 30 (1972), 62-64. MR0385990

[Sa2]     Sarason, D., Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207 (1975), 391-405. MR0377518

[Sa3]     Sarason, D., Function theory on the unit circle. Virginia Polytechnic Institute and State University, Blacksburg, VA, 1979. MR0521811

[SeSi]     Seidel, M. and Silbermann, B., Banach algebras of structured matrix sequences. Linear Algebra Appl. 430 (2009), 1243-1281. MR2489391

[Si1]     Simon, B., Orthogonal polynomials on the unit circle. Part 1: Classical theory. AMS Colloq. Publ. 54.1, AMS, Providence, RI, 2005. MR2105088

[Si2]     Simon, B., Orthogonal polynomials on the unit circle. Part 2: Spectral theory. AMS Colloq. Publ. 54.2, AMS, Providence, RI, 2005. MR2105089

[Si3]     Simon, B., Szegö’s theorem and its descendants. Spectral theory for L2 perturbations of orthogonal polynomials. Princeton Univ. Press, Princeton, NJ, 2011. MR2743058

[Ste]     Stein, E. M., Harmonic analysis. Princeton University Press. 1993. MR1232192

[Sto]     Stone, M. H., Linear transformations in Hilbert space and their applications to analysis. Coll. Publ. XV. American Math. Soc., 1932. MR1451877

[SzNF]     Sz.-Nagy, B. and Foias, C., Harmonic analysis of operators on Hilbert space, North-Holland, 1970 (2nd ed., with H. Bercovici and L. Kérchy, Springer Universitext, 2010). MR0275190

[TrVo1]     Treil, S. and Volberg, A., Wavelets and the angle between past and future. J. Functional Analysis 143 (1997), 269-308. MR1428818

[TrVo2]     Treil, S. and Volberg, A., Completely regular multivariate stochastic processes and the Muckenhoupt conditiion. Pacific J. Math. 190 (1999), 361-382. MR1722900

[Wh]     Whittle, P., On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix. Biometrika 50 (1963), 129-134. MR0161430

[Wi]     Wiener, N., Collected works, Volume III (ed. P. Masani), MIT Press, 1981. MR0652691

[WiAk]     Wiener, N. and Akutowicz, E. J., A factorization of positive Hermitian matrices, J. Math. Mech. 8 (1959), 111-120 (reprinted in [Wi] 264-273). MR0103388

[WiMa1]     Wiener, N. and Masani, P., The prediction theory of multivariate stochastic processes, I: The regularity condition. Acta Math. 98 (1957), 111-150 (reprinted in [Wi], 164-203). MR0097856

[WiMa2]     Wiener, N. and Masani, P., The prediction theory of multivariate stochastic processes, II: The linear predictor. Acta Math. 98 99 (1958), 93-137 (reprinted in [Wi] 164-203). MR0097859

[WiMa3]     Wiener, N. and Masani, P., On bivariate stationary processes and the factorization of matrix-valued functions. Th. Prob. Appl. 4 (1959), 300-308 (reprinted in [Wi] 255-263). MR0123368

[WigRo]     Wiggins, R. A. and Robinson, E. A., Recursive solution to the multichannel filtering problem. J. Geophys. Res. 70 (1965), 1885-1891. MR0183107




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