Summary
It is shown that formal Edgeworth expansions are valid for sums of weakly dependent random vectors. The error of approximation has ordero(n −(s−2)/2) if
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(i)
the moments of orders+1 are uniformly bounded
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(ii)
a conditional Cramér-condition holds
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(iii)
the random vectors can be approximated by other random vectors which satisfy a strong mixing condition and a Markov type condition.
The strong mixing coefficients in (iii) are decreasing at an exponential rate. The above conditions can easily be checked and are often satisfied when the sequence of random vectors is a Gaussian, or a Markov, or an autoregressive process. Explicit formulas are given for the distribution of finite Fourier transforms of a strictly stationary time series.
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References
Anderson, T.W.: The Statistical Analysis of Time Series. New York: Wiley 1971
Bhattacharya, R.N., Ranga Rao, R.: Normal Approximation and Asymptotic Expansions. New York: Wiley 1976
Bolthausen, E.: The Berry-Essen theorem for functionals of discrete Markov chains. Z.Wahrscheinlichkeitstheorie verw. Gebiete54, 59–74 (1980)
Bulinskii, A.V., Zhurbenko, I.G.: The central limit theorem for random fields. Dokl. Akad. Nauk SSSR 226. English translation in: Soviet Math. Dokl.17, 14–17 (1976)
Durbin, J.: Approximations for densities of sufficient estimators. Biometrika67, 311–333 (1980)
Götze, F., Hipp, C.: Asymptotic expansions in the central limit theorem under moment conditions. Z. Wahrscheinlichkeitstheorie verw. Gebiete42, 67–87 (1978)
Hannan, E.J.: Multiple Time Series. New York: Wiley 1970
Ibragimov, I.A.: Some limit theorems for stationary processes. Theor. Probability Appl.7, 349–382 (1962)
Ibragimov, I.A.: The central limit theorem for sums of functions of independent random variables and sums of the form∑f(2 k t). Theor. Probability Appl.12, 596–607 (1967)
Ibragimov, I.A.: On the spectrum of stationary Gaussian sequences satisfying the strong mixing condition. II. Sufficient conditions. Mixing rate. Theor. Probability Appl.15, 23–36 (1970)
Rosenblatt, M.: A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA42, 43–47 (1956)
Statulevicius, V.: Limit theorems for sums of random variables that are connected in a Markov chain. I, II, III. Litovsk. Mat. Sb. 9, 345–362;9, 635–672 (1969);10, 161–169 (1970)
Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Sympos. Math. Statist. Probability2, 583–602 (1972)
Sweeting, T.J.: Speeds of convergence for the multidimensional central limit theorem. Ann. Probab.5, 28–41 (1977)
Tikhomirov, A.N.: On the convergence rate in the central limit theorem for weakly dependent random variables. Theor. Probability Appl.25, 790–809 (1980)
Zhurbenko, I.G.: On strong estimates of mixed semiinvariants of random processes. Sibirskii Mat. Z.13, 293–308. English translation in: Siberian Math. J.13, 202–213 (1972)
Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications. Univ. California Press, Berkeley (1958)
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Götze, F., Hipp, C. Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrscheinlichkeitstheorie verw Gebiete 64, 211–239 (1983). https://doi.org/10.1007/BF01844607
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DOI: https://doi.org/10.1007/BF01844607