## Annals of Statistics

### Explicit representation of finite predictor coefficients and its applications

#### Abstract

We consider the finite-past predictor coefficients of stationary time series, and establish an explicit representation for them, in terms of the MA and AR coefficients. The proof is based on the alternate applications of projection operators associated with the infinite past and the infinite future. Applying the result to long memory processes, we give the rate of convergence of the finite predictor coefficients and prove an inequality of Baxter-type.

#### Article information

Source
Ann. Statist., Volume 34, Number 2 (2006), 973-993.

Dates
First available in Project Euclid: 27 June 2006

https://projecteuclid.org/euclid.aos/1151418248

Digital Object Identifier
doi:10.1214/009053606000000209

Mathematical Reviews number (MathSciNet)
MR2283400

Zentralblatt MATH identifier
1098.62120

#### Citation

Inoue, Akihiko; Kasahara, Yukio. Explicit representation of finite predictor coefficients and its applications. Ann. Statist. 34 (2006), no. 2, 973--993. doi:10.1214/009053606000000209. https://projecteuclid.org/euclid.aos/1151418248

#### References

• Baxter, G. (1962). An asymptotic result for the finite predictor. Math. Scand. 10 137–144.
• Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.
• Berk, K. N. (1974). Consistent autoregressive spectral estimates. Ann. Statist. 2 489–502.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation, 2nd ed. Cambridge Univ. Press.
• Bloomfield, P., Jewell, N. P. and Hayashi, E. (1983). Characterizations of completely nondeterministic stochastic processes. Pacific J. Math. 107 307–317.
• Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
• Cheng, R. and Pourahmadi, M. (1993). Baxter's inequality and convergence of finite predictors of multivariate stochastic processes. Probab. Theory Related Fields 95 115–124.
• Ginovian, M. S. (1999). Asymptotic behavior of the prediction error for stationary random sequences. J. Contemp. Math. Anal. 34 18–36.
• Granger, C. W. J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1 15–29.
• Grenander, U. and Szegö, G. (1958). Toeplitz Forms and Their Applications. Univ. California Press, Berkeley.
• Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952). Inequalities, 2nd ed. Cambridge Univ. Press.
• Helson, H. and Sarason, D. (1968). Past and future. Math. Scand. 21 5–16.
• Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165–176.
• Inoue, A. (1997). Regularly varying correlation functions and KMO-Langevin equations. Hokkaido Math. J. 26 457–482.
• Inoue, A. (2000). Asymptotics for the partial autocorrelation function of a stationary process. J. Anal. Math. 81 65–109.
• Inoue, A. (2001). What does the partial autocorrelation function look like for large lags? Preprint. Available at www.math.hokudai.ac.jp/~inoue/PT-01-123.pdf.
• Inoue, A. (2002). Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes. Ann. Appl. Probab. 12 1471–1491.
• Inoue, A. and Kasahara, Y. (2004). Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing. J. Multivariate Anal. 89 135–147.
• Kokoszka, P. S. and Taqqu, M. S. (1995). Fractional ARIMA with stable innovations. Stochastic Process. Appl. 60 19–47.
• Kolmogorov, A. N. (1941). Stationary sequences in Hilbert space. Bull. Math. Univ. Moscow 2 1–40. (In Russian.)
• Makagon, A. and Weron, A. (1976). Wold–Cramér concordance theorems for interpolation of $q$-variate stationary processes over locally compact Abelian groups. J. Multivariate Anal. 6 123–137.
• Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory. Wiley, New York.
• Rozanov, Y. A. (1967). Stationary Random Processes. Holden–Day, San Francisco.
• Salehi, H. (1979). Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes. Ann. Probab. 7 840–846.
• Sarason, D. (1978). Function theory on the unit circle. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, VA.
• Simon, B. (2005). Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory. Amer. Math. Soc., Providence, RI.
• Simon, B. (2005). Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory. Amer. Math. Soc., Providence, RI.