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Asymptotic expansions for central limit theorems for general linear stochastic processes. I: General theorems on rates of convergence. (English) Zbl 0419.60016


MSC:

60F05 Central limit and other weak theorems
60G20 Generalized stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
93E10 Estimation and detection in stochastic control theory
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[1] Brown, Moments of a stopping rule related to the central limit theorem, Ann. Math. Statist. 40 pp 1236– (1969) · Zbl 0212.21103 · doi:10.1214/aoms/1177697499
[2] Butzer , P. L. Gather , U. Asymptotic expansions for central limit theorems for general linear stochastic processes. II: Models of the general random noise and pulse train processes · Zbl 0419.60017
[3] Butzer, On the rate of approximation in the central limit theorem, J. Approximation Theory 13 pp 327– (1975) · Zbl 0298.60014 · doi:10.1016/0021-9045(75)90042-8
[4] Butzer, General theorems on rates of convergence in distribution of random variables, I: General limit theorems; II: Applications to the stable limit law and weak law of large numbers, J. Multivariate Anal. 8 pp 181– (1978) · Zbl 0383.60024 · doi:10.1016/0047-259X(78)90071-4
[5] Butzer, Fourier Analysis and Approximation 1 (1971) · doi:10.1007/978-3-0348-7448-9
[6] Chanda, Asymptotic expansions for a class of distribution functions, Ann. Math. Statist. 34 pp 1302– (1963) · Zbl 0237.60014 · doi:10.1214/aoms/1177703865
[7] Cramér, Random Variables and Probability Distributions (1962)
[8] Doob, Stochastic Processes (1953)
[9] Endow, A central limit theorem for the random noise process, Keio Engrg. Rep. 25 pp 19– (1972)
[10] Endow , Y. Some limit theorems for the general random noise process 1973 36 41 · Zbl 0385.60041
[11] Esseen, Fourier analysis of distribution functions, a mathematical study of the Laplace Gaussian law, Acta Math. 77 pp 3– (1944)
[12] Franks, Signal Theory (1969)
[13] Gihman, The theory of Stochastic Processes I (1974) · Zbl 0291.60019 · doi:10.1007/978-3-642-61943-4
[14] Gnedenko, Grenzverteilungen von Summen unabhängiger Zufallsgrößen (1960)
[15] Grenander, Statistical Analysis of Stationary Time Series (1956)
[16] Honda, On the central limit theorem for a class of general linear processes, Keio Engrg, Rep. 26 pp 27– (1973)
[17] Ibragimov, Independent and Stationary Sequences of Random Variables (1971)
[18] Kampé de Feriet, Correlations and spectra for nonstationary random functions, Math. Comp. 16 pp 1– (1962) · doi:10.1090/S0025-5718-1962-0137265-4
[19] Kawata, On the stochastic process of random noise, Kodai Math, Sem. Reports 7 pp 23– (1955) · Zbl 0066.11502
[20] Kawata, On a class of linear processes. Proc. Second Japan-U. S. S. R. Symposium on Probability Theory and Statistics pp 193– (1972) · Zbl 0283.60047
[21] Kawata, Nonstationary stochastic processes, Keio Math. Sem. Reports 1 pp 1– (1973) · Zbl 0045.17903 · doi:10.2996/kmj/1138833371
[22] Kawata, Some analytic properties of random noise processes, Keio Engr. Rep. 27 pp 73– (1974) · Zbl 0401.60040
[23] Lucky, Principles of Data Communication (1968)
[24] Lugannani, On a class of stochastic processes which are closed under linear transformations, Information and Control 10 pp 1– (1967) · Zbl 0157.24901 · doi:10.1016/S0019-9958(67)90022-8
[25] Lugannani, The central limit theorem for a class of stochastic processes, J. Math. Anal. Appl. 24 pp 25– (1968) · Zbl 0247.60019 · doi:10.1016/0022-247X(68)90047-4
[26] Lugannani, Convergence properties of the sample mean and sample correlation for a class of pulse trains, SIAM J. Appl. Math. 21 pp 1– (1971) · Zbl 0204.52904 · doi:10.1137/0121001
[27] Middleton, An Introduction to Statistical Communication Theory (1960) · Zbl 0111.32501
[28] Parzen, Statistical analysis of asymptotically stationary time series, Bull. Inst. Intern. Statis. 39 pp 87– (1962) · Zbl 0122.37201
[29] Petrov, On local limit theorems for sums of independent random variables, Theor. Prob. Appl. 9 pp 312– (1964) · Zbl 0146.38003 · doi:10.1137/1109044
[30] Petrov, Sums of Independent Random Variables (1975) · doi:10.1007/978-3-642-65809-9
[31] Pierre, On the independence of linear functionals of linear processes, SIAM J. Appl. Math. 17 pp 624– (1969) · Zbl 0185.44603 · doi:10.1137/0117060
[32] Pierre, Central limit theorems for conditionally linear random processes, SIAM J. Appl. Math. 20 pp 449– (1971) · Zbl 0218.60027 · doi:10.1137/0120048
[33] Rice, Mathematical analysis of random noise, Bell System Techn. J. pp 1– (1944)
[34] Rozanov, Stationary Random Processes (1967)
[35] Statulevicius , V. A. On the sharpening of limit theorems for weakly dependent random variables, Trudy IV 1962
[36] Sun, A central limit theorem for nonlinear functions of a normal stationary process, J. Math. Mech. 12 pp 945– (1963)
[37] Thomas, Introduction to Statistical Communication Theory (1969) · Zbl 0202.17801
[38] Wolff, IEEE Trans. Information Theory IT-13, in: On probability distributions for filtered white noise pp 481– (1967)
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