Summary
For an r>2 and a finite \(K,E\left| {\sum\limits_{j = 1}^n {X_j } } \right|^r \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } Kn^{r/2}\) (all n≧1) is obtained for a strictly stationary strong mixing sequence {X j }. The convergence of rth(r>2) absolute moments in the central limit theorem for stationary φ-mixing and strong mixing sequences is also studied.
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von Bahr, Bengt: On the convergence of moments in the central limit theorem. Ann. Math. Statist. 36, 808–818 (1965)
Bernstein, S.: Quelques remarques sur le théorème limite Liapounoff. Dokl. Akad. Nauk SSSR 24, 3–8 (1939)
Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968
Brillinger, D.: A note on the rate of convergence of mean. Biometrika 49, 574–576 (1962)
Brown, B.M.: Moments of a stopping rule related to the central limit theorem. Ann. Math. Statist. 40, 1236–1249 (1969)
Brown, B.M.: Characteristic functions, moments and the central limit theorem. Ann. Math. Statist. 41, 658–664 (1970)
Davydov, Yu.A.: Convergence of distributions generated by stationary stochastic processes. Theor. Probability Appl. 13, 691–696 (1968)
Doob, J.L.: Stochastic Processes. New York: Wiley 1953
Hall, P.: The convergence of moments in the martingale central limit theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 44, 253–260 (1978)
Ibragimov, I.A.: Some limit theorems for stationary processes. Theor. Probability Appl. 7, 349–382 (1962)
Ibragimov, I.A., Linnik, Yu.V.: Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff 1971
Sen, P.K.: Weak convergence of multidimensional empirical processes for stationary φ-mixing processes. Ann. Probability 2, 147–154 (1974)
Serfling, R.J.: Moment inequalities for the maximum cumulative sum. Ann. Math. Statist. 41, 1227–1234 (1970)
Serfling, R.J.: Convergence properties of S n under moment restrictions. Ann. Math. Statist. 41, 1235–1248 (1970)
Stout, W.F.: Almost Sure Convergence. New York: Academic Press 1974
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Yokoyama, R. Moment bounds for stationary mixing sequences. Z. Wahrscheinlichkeitstheorie verw Gebiete 52, 45–57 (1980). https://doi.org/10.1007/BF00534186
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DOI: https://doi.org/10.1007/BF00534186