Abstract
Let X(t) (t = 0, ± 1,... ) be a zero mean, r vector-valued, strictly stationary time series satisfying a particular assumption about the near-independence of widely separated values.
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References
ANDERSON, T. W. (1958). Introduction. to Multivariate Statistical Analysis. New York: Wiley.
ANDERSON, T. W. & WALKER, A. M. (1964). On the asymptotic distribution of the autocorrelations of a sample from a linear stochastic process. Ann. Math. Statist. 35, 1296–1303.
BARTLETT, M. S. (1946). On the theoretical specification of sampling properties of auto-correlated time series. J. R. Statist. Soc. Suppl. 8, 27–41.
BARTLETT, M. S. (1950). Periodogram analysis and continuous spectra. Biometrika 37, 1–16.
BARTLETT, M. S. (1966). An Introduction to Stochastic Processes, 2nd edition. Cambridge University Press.
BLACKMAN, R. B. & TUKEY,.T. W. (1958). The Measurement of Power Spectra. New York: Dover.
BRILLINGER, D. R. (1965). An introduction to polyspectra. Ann. Math. Statist. 36, 1351–74.
BRILLINGER,D. R. (1968). Estimation of the cross-spectrum of a stationary bivariate Gaussian process from its zeros. J. R. Statist. Soc. B 30, 145–59.
BRILLINGER, D. R. & ROSENBLATT, M. (1967). Asymptotic theory of estimates of k-th order spectra. In Advanced Seminar on Spectral Analysis of Time Series (ed, B. Harris), pp. 153–88. New York: Wiley.
EDWARDS, R. Eo (1967). Fourier Series, vol. 1. New York: Holt, Rinehart and Winston.
GOODMAN, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann. Math. Statist. 34, 152–77.
GRENANDER, U. & ROSENBLATT, M. (1957). Statistical Analysis of Stationary Time Series. New York: Wiley.
HANNAN, Eo.T. (1960). Time Series Analysis. London: Methuen.
HANNAN, E..T. (1967). The estimation of a lagged regression relation. Biometrika 54, 409–18.
IBRAGIMOV, I. A. (1963). On estimation of the spectral function of a stationary Gaussian process. Theory Probab. Appl. 8, 366–401.
KAWATA, T. (1959). Some convergence theorems for stationary stochastic processes. Ann. Math. Statist. 30, 1192–214.
LAMPERTI,.T. (1962). On convergence of stochastic processes. Trans. Amer. Math. Soc. 104, 430–5.
LEONOV, V. P. & SHIRYAEV, A. N. (1959). On a method of calculation of semi-invariants. Theory Probab. Appl. 4, 319–29.
LOMNICKI, Z. A. & ZAREMBA, S. K. (1957). On some moments and distributions occurring in the theory of linear stochastic processes. Part 1. Mh, Math. 61, 318–58.
LOMNICKI, Z. A. & ZAREMBA, S. K. (1959). On some moments and distributions occurring in the theory of linear stochastic processes. Part II. Mh. Math. 63, 128–68.
MALEVICH, T. L. (1964). The asymptotic behavior ofan estimate for the spectral function ofa stationary Gaussian process. Theory Probab. Appl. 9, 350–3.
MALEVICH, T. L. (1965). Some properties of the estimators of the spectrum of a stationary process. Theory Probab. Appl. 10, 457–65.
OLSHEN, R. A. (1967). Asymptotic properties of the periodogram of a discrete stationary process. J. Appl. Prob. 4, 508–28.
PARZEN, E. (1957a). A central limit theorem for multilinear stochastic processes. Ann. Math. Statist. 28,252-5.
PARZEN, Eo (1957 b). On consistent estimates of the spectrum of a stationary time series. Ann. Math, Statist. 28, 329–48.
PARZEN, Eo (1961). An approach to time series analysis. Ann. Math. Statist. 32, 951–89.
RAO, S. T. (1967). On the cross periodogram of a stationary Gaussian vcctor process. Ann. Math. Statist. 38, 593–7.
ROSENBLATT, M. (1959). Statistical analysis of stochastic processes with stationary residuals. In Studies in Probability and Statistics (ed. U. Grenander), pp. 246–75. New York: Wiley.
ROSENBLATT, M. (1962). Asymptotic behavior of eigenvalues of Toeplitz forms. J. Math, Mech, 11, 941 –50.
SLUTSKY, E. E. (1934). Alcunc applicazioni dei coefficienti di Fourier all 'analisi delle funzioni aleat orie stazionarie. Gion. 1st. ltal. Attuari 5, 1–50.
TuKEY, J. W. (1967). An introduction to the calculations of numerical spectrum analysis. In Advanced S eminar on Spectral Analysis of Time S eries (ed. B. Harris), pp. 25–46. New York: Wiley.
WARBA, G. (1968). On the distribution of some statistics useful in the analysis of jointly stationary time series. Ann. Math. Statist. 39, 1849–62.
WALKER, A. M. (1954). The asymptotic distribution of serial correlation coefficients for autoregressive p ro cesses with dependent residuals, Pro c, Camb. Phil. Soc. 50, 60–4.
WALKER, A. M. (1965). Some asymptotic results for the pcriodogram of a stationary time series, J. Aust. Math. Soc. 5, 107–28.
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Brillinger, D.R. (2012). Asymptotic properties of spectral estimates of second order. In: Guttorp, P., Brillinger, D. (eds) Selected Works of David Brillinger. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1344-8_12
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