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The intersection of past and future for multivariate stationary processes


Authors: Akihiko Inoue, Yukio Kasahara and Mohsen Pourahmadi
Journal: Proc. Amer. Math. Soc. 144 (2016), 1779-1786
MSC (2010): Primary 60G10; Secondary 62M10, 60G25
DOI: https://doi.org/10.1090/proc/12869
Published electronically: September 9, 2015
MathSciNet review: 3451253
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider an intersection of past and future property of multivariate stationary processes which is the key to deriving various representation theorems for their linear predictor coefficient matrices. We extend useful spectral characterizations for this property from univariate processes to multivariate processes.


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  • [BIK] N. H. Bingham, Akihiko Inoue, and Yukio Kasahara, An explicit representation of Verblunsky coefficients, Statist. Probab. Lett. 82 (2012), no. 2, 403-410. MR 2875229, https://doi.org/10.1016/j.spl.2011.11.004
  • [BJH] Peter Bloomfield, Nicholas P. Jewell, and Eric Hayashi, Characterizations of completely nondeterministic stochastic processes, Pacific J. Math. 107 (1983), no. 2, 307-317. MR 705750 (85g:60053)
  • [H] Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982. Encyclopedia of Mathematics and its Applications, 17. MR 675952 (84e:47001)
  • [HL] Henry Helson and David Lowdenslager, Prediction theory and Fourier series in several variables. II, Acta Math. 106 (1961), 175-213. MR 0176287 (31 #562)
  • [I1] Akihiko Inoue, Asymptotics for the partial autocorrelation function of a stationary process, J. Anal. Math. 81 (2000), 65-109. MR 1785278 (2001e:60075), https://doi.org/10.1007/BF02788986
  • [I2] Akihiko Inoue, Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes, Ann. Appl. Probab. 12 (2002), no. 4, 1471-1491. MR 1936600 (2003i:62143), https://doi.org/10.1214/aoap/1037125870
  • [I3] Akihiko Inoue, AR and MA representation of partial autocorrelation functions, with applications, Probab. Theory Related Fields 140 (2008), no. 3-4, 523-551. MR 2365483 (2009b:62176), https://doi.org/10.1007/s00440-007-0074-1
  • [IK1] Akihiko Inoue and Yukio Kasahara, Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing, J. Multivariate Anal. 89 (2004), no. 1, 135-147. MR 2041213 (2005a:62197), https://doi.org/10.1016/S0047-259X(02)00027-1
  • [IK2] Akihiko Inoue and Yukio Kasahara, Explicit representation of finite predictor coefficients and its applications, Ann. Statist. 34 (2006), no. 2, 973-993. MR 2283400 (2008c:60036), https://doi.org/10.1214/009053606000000209
  • [KK] V. E. Katsnelson and B. Kirstein, On the theory of matrix-valued functions belonging to the Smirnov class, Topics in interpolation theory (Leipzig, 1994) Oper. Theory Adv. Appl., vol. 95, Birkhäuser, Basel, 1997, pp. 299-350. MR 1473261 (98m:47016)
  • [KB] Yukio Kasahara and Nicholas H. Bingham, Verblunsky coefficients and Nehari sequences, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1363-1378. MR 3145734, https://doi.org/10.1090/S0002-9947-2013-05874-6
  • [LM] N. Levinson and H. P. McKean Jr., Weighted trigonometrical approximation on $ R^{1}$ with application to the germ field of a stationary Gaussian noise, Acta Math. 112 (1964), 99-143. MR 0163111 (29 #414)
  • [P] Mohsen Pourahmadi, Foundations of time series analysis and prediction theory, Wiley Series in Probability and Statistics: Applied Probability and Statistics, Wiley-Interscience, New York, 2001. MR 1849562 (2002f:62090)
  • [RR] Marvin Rosenblum and James Rovnyak, Topics in Hardy classes and univalent functions, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1307384 (97a:30047)
  • [R] Yu. A. Rozanov, Stationary random processes, Translated from the Russian by A. Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1967. MR 0214134 (35 #4985)
  • [S] Donald Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19-23, 1978. MR 521811 (80d:30035)

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Additional Information

Akihiko Inoue
Affiliation: Department of Mathematics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Email: inoue100@hiroshima-u.ac.jp

Yukio Kasahara
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0811, Japan
Email: y-kasa@math.sci.hokudai.ac.jp

Mohsen Pourahmadi
Affiliation: Department of Statistics, Texas A&M University, College Station, Texas 77843
Email: pourahm@stat.tamu.edu

DOI: https://doi.org/10.1090/proc/12869
Received by editor(s): December 31, 2014
Received by editor(s) in revised form: April 17, 2015
Published electronically: September 9, 2015
Additional Notes: The third author was supported by NFS grant DMS-1309586.
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society

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