Multivariate prediction and matrix Szegő theory. (English) Zbl 1285.60038

Bingham, N. H.

Probab. Surv. 9, 325-339 (2012).

The paper is a multivariate sequel to a survey article on Szegő’s theorem and orthogonal polynomials on the unit circle (OPUC), published by N. H. Bingham in [Probab. Surv. 9, 325–339 (2012; Zbl 1285.60038)]. Similarly to the scalar version, the author starts with the Kolmogorov isomorphism theorem, which now relates an \(l\)-dimensional complex-valued zero-mean covariance stationary stochastic process \( X = (X_n : n\in{\mathbb Z})\) to an \(l\)-dimensional process \(Y\) on the unit circle \(\mathbb T\) with orthogonal increments, and an \(l\times l\) matrix-valued probability measure \(\mu\) on \(\mathbb T\). In the multivariate setting, the Verblunsky coefficients, which have been studied by D. Damanik et al. [Surv. Approx. Theory 4, 1–85 (2008; Zbl 1193.42097)], are \(l\times l\) matrices \({\alpha}_n\) such that \(\| {\alpha}_n \| < 1\), where \(\| \cdot \|\) denotes the Euclidean norm. Their multivariate Verblunsky theorem establishes a bijection between the sequences \(\alpha = ({\alpha}_n)^{\infty}_{n=1}\) of the Verblunsky coefficients and the non-trivial \(l\times l\) matrix-valued probability measures \(\mu\) on \(\mathbb T\).
The rest of the paper reports on multivariate extensions of various conditions involving prediction theory and Szegő’s theorem, whose scalar versions are dealt with in the cited paper by the author. The article is accompanied by an exhaustive bibliography of the subject.

Reviewer: Ryszard Doman (Poznań)

MSC:

60G10	Stationary stochastic processes
60G25	Prediction theory (aspects of stochastic processes)
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators

Keywords:

vector-valued stationary process; multivariate prediction; multivariate orthogonal polynomials on the unit circle; Verblunsky’s theorem; Szegő’s theorem

Citations:

Zbl 1193.42097; Zbl 1285.60038

References:

[1]	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. · Zbl 0179.46701 · doi:10.1007/BF01075679
[2]	Arov, D. Z. and Dym, H., Matricial Nehari problems, \(J\)-inner functions and the Muckenhoupt condition. J. Funct. Anal. 181 (2001), 227-299. · Zbl 0980.47014 · doi:10.1006/jfan.2000.3725
[3]	Arov, D. Z. and Dym, H., Criteria for the strong regularity of \(J\)-inner functions and \(\gamma\)-generating functions. J. Math. Anal. Appl. 280 (2003), 387-399. · Zbl 1044.47012 · doi:10.1016/S0022-247X(03)00067-2
[4]	Arov, D. Z. and Dym, H., \(J\)-contractive matrix valued functions and related topics , Encycl. Math. Appl. 116, Cambridge Univ. Press, Cambridge, 2008. · Zbl 1159.30001
[5]	Barclay, S. J., Continuity of the spectral factorization maping. J. London Math. Soc. 70 (2004), 763-779. · Zbl 1171.47301 · doi:10.1112/S0024610704005575
[6]	Barclay, S. J., A solution of the Douglas-Rudin problem for matrix-valued functions. Proc. London Math. Soc. 99 (2009), 757-786. · Zbl 1192.47012 · doi:10.1112/plms/pdp017
[7]	Basor, E. L. and Widom, H., On a Toeplitz determinant identity of Borodin and Okounkov. Integral Equations and Operator Theory 37 (2000), 397-401. · Zbl 0967.47020 · doi:10.1007/BF01192828
[8]	Baxter, G., A convergence equivalence related to polynomials orthogonal on the unit circle. Trans. Amer. Math. Soc. 99 (1961), 471-487. · Zbl 0116.35705 · doi:10.2307/1993557
[9]	Baxter, G., An asymptotic result for the finite predictor. Math. Scand. 10 (1962), 137-144. · Zbl 0112.09401
[10]	Baxter, G., A norm inequality for a “finite-section” Wiener-Hopf equation. Illinois J. Math. 7 (1963), 97-103. · Zbl 0113.09101
[11]	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). · Zbl 0033.37701 · doi:10.1007/BF02395019
[12]	Bingham, N. H., Szegö’s theorem and its probabilistic descendants. Probability Surveys 9 (2012), 287-324. · Zbl 1285.60037
[13]	Bingham, N. H. and Fry, J. M., Regression: Linear models in statistics . SUMS (Springer Undergraduate Mathematics Series), 2010. · Zbl 1245.62085 · doi:10.1007/978-1-84882-969-5
[14]	Bingham, N. H., Fry, J. M. and Kiesel, R., Multivariate elliptic processes. Statistica Neerlandica 64 (2010), 352-366. · doi:10.1111/j.1467-9574.2010.00465.x
[15]	Bingham, N. H., Inoue, A. and Kasahara, Y., An explicit representation of Verblunsky coefficients. Statistics and Probability Letters 82 .2 (2012), 403-410. · Zbl 1237.62111
[16]	Bloomfield, P., Jewell, N. P. and Hayashi, E., Characterization of completely non-deterministic stochastic processes. Pacific J. Math. 107 (1983), 307-317. · Zbl 0472.60041 · doi:10.2140/pjm.1983.107.307
[17]	Böttcher, A., One more proof of the Borodin-Okounkiv formula for Toeplitz determinants. Integral Equations and Operator Theory 41 (2001), 13 -125. · Zbl 0992.47013 · doi:10.1007/BF01202535
[18]	Böttcher, A., Featured review of the Borodin-Okounkov and Basor-Widom papers. Mathematical Reviews 1790118/9 (2001g:47042a,b).
[19]	Böttcher, A., On the determinant formulas by Borodin, Okounkov, Baik, Deift and Rains. Operator Th. Adv. Appl. 135, 91-99, Birkhäuser, Basel, 2002. · Zbl 1040.47016
[20]	Chanzy, J., Théorèmes-limite de Szëgo dans le cas matriciel. Proc. Japan Acad. A 82 (2006), 113-116. · Zbl 1182.47028 · doi:10.3792/pjaa.82.113
[21]	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. · Zbl 1138.47023 · doi:10.5802/ambp.216
[22]	Cheng, R. and Pourahmadi, M., Baxter’s inequality and convergence of predictors of multivariate stochastic processes. Probab. Th. Rel. Fields 95 (1993), 115-124. · Zbl 0792.60031 · doi:10.1007/BF01197341
[23]	Curtain, R. F. and Zwart, H., An introduction to infinite-dimensional linear systems . Springer, 1995. · Zbl 0839.93001
[24]	Damanik, D., Pushnitski, A. and Simon, B., The analytic theory of matrix orthogonal polynomials. Surveys in Approximation Theory 4 (2008), 1-85. · Zbl 1193.42097
[25]	Derevyagin, M., Holtz, O., Khrushchev, S. and Tyaglov, M., Szegö’s theorem for matrix orthogonal polynomials. arXiv: · Zbl 1276.33013 · doi:10.1016/j.jat.2012.05.003
[26]	Devinatz, A., The factorization of operator-valued functions. Ann. Math. 73 (1961), 458-495. · Zbl 0098.08402 · doi:10.2307/1970313
[27]	Dubovoj, V. K., Fritzsche, B. and Kirstein, B., Matricial version of the classical Schur problem . Teubner, Stuttgart, 1992. · Zbl 0754.47013
[28]	Ephremidze, L., Janashia, G. and Lagvilava, E., An analytic proof of the matrix spectral factorization theorem. Georgian Math. J. 15 (2008), 241-249. · Zbl 1160.47017
[29]	Ephremidze, L. and Lagvilada, E., Remark on outer analytic matrix functions. Proc. A. Razmadze Math. Inst. 152 (2010), 29-32. · Zbl 1196.30047
[30]	Frazho, A. E. and Bhosri, W., An operator perspective on signals and systems . Operator Theory: advances and Applications 204 , Birkhäuser, 2010. · Zbl 1218.93003 · doi:10.1007/978-3-0346-0292-1
[31]	Foias, C. and Frazho, A. E., The commutant lifting approach to interpolation problems . Operator Theory: Advances and Applications 44 , Birkhäuser, 1990. · Zbl 0718.47010
[32]	Grenander, U. and Szegö, G., Toeplitz forms and their applications . Univ. California Press, Berkeley, CA, 1958. · Zbl 0080.09501
[33]	Halmos, P. R., Shifts on Hilbert spaces. J. reine ang. Math. 208 (1961), 102-112. · Zbl 0107.09802 · doi:10.1515/crll.1961.208.102
[34]	Hannan, E. J., Multiple time series , Wiley, 1970 · Zbl 0211.49804
[35]	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. · Zbl 0082.28201 · doi:10.1007/BF02392425
[36]	Helson, H. and Sarason, D., Past and future. Math. Scand 21 (1967), 5-16. · Zbl 0241.60029
[37]	H. Helson and G. Szegö, A problem in prediction theory. Acta Mat. Pura Appl. 51 (1960), 107-138. · Zbl 0178.50002 · doi:10.1007/BF02410947
[38]	Hille, E. and Phillips, R. S., Functional anaysis and semigroups . Colloq. Publ. 31, American Math. Soc., 1957. · Zbl 0078.10004
[39]	Ibragimov, I. A. and Rozanov, Yu. A., Gaussian random processes . Springer, 1978. · Zbl 0392.60037
[40]	Inoue, A., Asymptotics for the partial autocorrelation function of a stationary process. J. Analyse Math. 81 (2000), 65-109. · Zbl 0971.60039 · doi:10.1007/BF02788986
[41]	Inoue, A., Asymptotic behaviour for partial autocorrelation functions of fractional ARIMA processes. Ann. Appl. Prob. 12 (2002), 1471-1491. · Zbl 1073.62553 · doi:10.1214/aoap/1037125870
[42]	Inoue, A., AR and MA presentations of partial autocorrelation functions with applications. Prob. Th. Rel. Fields 140 (2008), 523-551. · Zbl 1138.62049 · doi:10.1007/s00440-007-0074-1
[43]	Inoue, A. and Kasahara, Y., Partial autocorrelation functions of the fractional ARIMA process. J. Multivariate Analysis 89 (2004), 135-147. · Zbl 1036.62079 · doi:10.1016/S0047-259X(02)00027-1
[44]	Inoue, A. and Kasahara, Y., Explicit representation of finite predictor coefficients and its applications. Ann. Statist. 34 (2006), 973-993. · Zbl 1098.62120 · doi:10.1214/009053606000000209
[45]	Janashia, G., Lagvilada, E. and Ephremidze, L., A new method of matrix spectral factorization. IEEE Trans. Info. Th. 57 (2011), 2318-2326. · Zbl 1367.65064 · doi:10.1109/TIT.2011.2112233
[46]	Kallenberg, O., Foundations of modern probability , 2nd ed., Springer, 2002. · Zbl 0996.60001
[47]	Kasahara, Y. and Bingham, N. H., Verblunsky coefficients and Nehari sequences. Trans. Amer. Math. Soc. , · Zbl 1284.42080
[48]	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). · Zbl 0063.03291
[49]	Lax, P. D., Translation-invariant subspaces. Acta Math. 101 (1959), 163-178. · Zbl 0085.09102 · doi:10.1007/BF02559553
[50]	Lax, P. D., On the regularity of spectral densities. Th. Probab. Appl. 8 (1963), 337-340. · Zbl 0122.13502
[51]	Lax, P. D., On the factorization of matrix-valued functions. Comm. Pure Appl. Math. 29 (1976), 683-688. · Zbl 0326.47033 · doi:10.1002/cpa.3160290609
[52]	Levinson, N. and McKean, H. P., Weighted trigonometrical approximation on \(R^{1}\) 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). · Zbl 0126.13901 · doi:10.1007/BF02391767
[53]	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. · Zbl 0927.60052 · doi:10.1090/S0002-9939-98-04245-2
[54]	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. · Zbl 0676.60051 · doi:10.1007/BFb0083393
[55]	Makagon, A. and Weron, A., \(q\)-variate minimal stationary processes. Studia Math. 59 (1976), 41-52. · Zbl 0412.60013
[56]	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. · Zbl 0095.12501
[57]	Masani, P., The prediction theory of multivariate stochastic processes, III. Acta Math. 104 (1960), 141-162. · Zbl 0096.11505 · doi:10.1007/BF02547188
[58]	Masani, P., Shift-invariant spaces and prediction theory. Acta Math. 107 (1962), 275-290. · Zbl 0113.12304 · doi:10.1007/BF02545791
[59]	Masani, P., Recent trends in multivariate prediction theory. Multivariate Analysis (Proc. Int. Symp., Dayton OH) 351-382, Academic Press, 1966. · Zbl 0216.46706
[60]	Masani, P., Comments on the prediction-theoretic papers, 276-305 in [Wi].
[61]	Matveev, R. F., Regularity of multi-dimensional stochastic processes with discrete time. Dokl. Akad. Nauk SSSR 126 (1959), 713-715. · Zbl 0089.34006
[62]	Tao Mei, Operator-valued Hardy spaces . Memoirs Amer. Math. Soc. vol. 188 no. 881, 2007. · Zbl 1138.46038
[63]	Morf, M., Vieira, A. and Kailath, T., Covariance characterization by partial autocorrelation matrices. Ann. Statist. 6 (1978), 643-678. · Zbl 0413.62072 · doi:10.1214/aos/1176344208
[64]	Nikolskii, N. K., Treatise on the shifgt operator: Spectral function theory . Grundl. math. Wiss. 237 , Springer, 1986.
[65]	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. · Zbl 1331.35083
[66]	Peller, V. V., Hankel operators and multivariate stochastic processes. Proc. Symp. Pure Math. 51, Part 1, 357-371, AMS, Providence, RI, 1990. · Zbl 0712.47020
[67]	Peller, V. V., Factorization and approximation problems for matrix functions. J. American Math. soc. 11 (1998), 751-770. · Zbl 0903.47016 · doi:10.1090/S0894-0347-98-00274-4
[68]	Peller, V. V., Hankel operators and their applications , Springer, 2003. · Zbl 1030.47002
[69]	Pourahmadi, M., A matricial extension of the Helson-Szegö theorem and its application in multivariate prediction. J. Multivariate Analysis 16 (1985), 265-275. · Zbl 0583.62086 · doi:10.1016/0047-259X(85)90039-9
[70]	Pourahmadi, M., Joint mean-covariance models with applications to longitudinal data. Unconstrained parametrization. Biometrika 86 (1999), 677-690. · Zbl 0949.62066 · doi:10.1093/biomet/86.3.677
[71]	Pourahmadi, M., Foundations of time series analysis and prediction theory . Wiley, 2001. · Zbl 0982.62074
[72]	Reinsel, G. C., Elements of multivariate time series analysis , Springer, 1997. · Zbl 0873.62086
[73]	Riesz, F. and Sz.-Nagy, B., Leçons d’analyse fonctionnelle , 2nd ed., Akad. Kiadó, 1953. · Zbl 0051.08403
[74]	Rosenblum, M. and Rovnyak, J., Hardy classes and operator theory , Dover, New York, 1997 (1st ed. Oxford University Press, 1985). · Zbl 0586.47020
[75]	Rozanov, Yu. A., Spectral properties of multivariate stationary processes and boundary properties of analytic functions. Th. Probab. Appl. 5 (1960), 362-376. · Zbl 0094.12201
[76]	Rozanov, Yu. A., Stationary random processes , Holden Day, San Francisco, CA, 1967. · Zbl 0152.16302
[77]	Sarason, D., An addendum to “Past and future”. Math. Scand. 30 (1972), 62-64. · Zbl 0266.60023
[78]	Sarason, D., Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207 (1975), 391-405. · Zbl 0319.42006 · doi:10.2307/1997184
[79]	Sarason, D., Function theory on the unit circle . Virginia Polytechnic Institute and State University, Blacksburg, VA, 1979. · Zbl 0398.30027
[80]	Seidel, M. and Silbermann, B., Banach algebras of structured matrix sequences. Linear Algebra Appl. 430 (2009), 1243-1281. · Zbl 1161.47010 · doi:10.1016/j.laa.2008.10.021
[81]	Simon, B., Orthogonal polynomials on the unit circle. Part 1: Classical theory . AMS Colloq. Publ. 54.1, AMS, Providence, RI, 2005. · Zbl 1082.42020
[82]	Simon, B., Orthogonal polynomials on the unit circle. Part 2: Spectral theory . AMS Colloq. Publ. 54.2, AMS, Providence, RI, 2005. · Zbl 1082.42021
[83]	Simon, B., Szegö’s theorem and its descendants. Spectral theory for \(L_{2}\) perturbations of orthogonal polynomials . Princeton Univ. Press, Princeton, NJ, 2011. · Zbl 1230.33001
[84]	Stein, E. M., Harmonic analysis . Princeton University Press. 1993. · Zbl 0821.42001
[85]	Stone, M. H., Linear transformations in Hilbert space and their applications to analysis . Coll. Publ. XV. American Math. Soc., 1932. · Zbl 0005.40003
[86]	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). · Zbl 0201.45003
[87]	Treil, S. and Volberg, A., Wavelets and the angle between past and future. J. Functional Analysis 143 (1997), 269-308. · Zbl 0876.42027 · doi:10.1006/jfan.1996.2986
[88]	Treil, S. and Volberg, A., Completely regular multivariate stochastic processes and the Muckenhoupt conditiion. Pacific J. Math. 190 (1999), 361-382. · Zbl 1006.60027 · doi:10.2140/pjm.1999.190.361
[89]	Whittle, P., On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix. Biometrika 50 (1963), 129-134. · Zbl 0129.11304 · doi:10.1093/biomet/50.1-2.129
[90]	Wiener, N., Collected works, Volume III (ed. P. Masani), MIT Press, 1981.
[91]	Wiener, N. and Akutowicz, E. J., A factorization of positive Hermitian matrices, J. Math. Mech. 8 (1959), 111-120 (reprinted in [Wi] 264-273). · Zbl 0082.28103
[92]	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). · Zbl 0080.13002 · doi:10.1007/BF02404472
[93]	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). · doi:10.1007/BF02392423
[94]	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). · Zbl 0089.34003 · doi:10.1137/1104029
[95]	Wiggins, R. A. and Robinson, E. A., Recursive solution to the multichannel filtering problem. J. Geophys. Res. 70 (1965), 1885-1891. · doi:10.1029/JZ070i008p01885