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Almost sure invariance principles for mixing sequences of random variables. (English) Zbl 0793.60038

Let \((X_ n, n\geq 1)\) be a stationary mixing sequence of random variables with mean zero and finite variance; \(\rho(n)\) and \(\varphi(n)\) denote coefficients of \(\rho\)-mixing and uniform mixing, respectively. As usual, \(S_ n=X_ 1+\cdots+X_ n\), \(\sigma^ 2_ n=ES^ 2_ n\), \(n\geq 1\). It is established that if \(\sigma^ 2_ n@>>n\to\infty>\infty\) and the mixing rate satisfies \(\sum_ n\varphi^{1/2}(2^ n)<\infty\) or \(\rho(n)=O(\text{log}^{-r})\) for some \(r>1\), then \((S_ n, n\geq 1)\) could be redefined on a richer probability space on which there exists a standard Wiener process \((W(t), t\geq 0)\) such that \(S_ n-W(\sigma^ 2_ n)=o(\sigma_ n\sqrt{\text{log log} n})\) almost surely as \(n\to\infty\).

MSC:

60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
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[1] Berkes, I.; Philipp, W., Approximation theorems for independent and weakly dependent random variables, Ann. Probab., 7, 29-54 (1979) · Zbl 0392.60024
[2] Chow, Y. S., Local convergence of martingale and the law of large numbers, Ann. Math. Statist., 36, 552-558 (1965) · Zbl 0134.34003
[3] Csörgo&#x030B;, M.; Révész, P., Strong Approximations in Probability and Statistics (1981), Academic Press: Academic Press New York · Zbl 0539.60029
[4] Dabrowski, A. R., A note on a theorem of Berkes and Philipp for dependent sequences, Statist. Probab. Lett., 1, 53-55 (1982) · Zbl 0496.60031
[5] Einmahl, U., Strong invariance principles for partial sums of independent random vectors, Ann. Probab., 15, 1419-1440 (1987) · Zbl 0637.60041
[6] Hall, P.; Heyde, C. C., Martingale Limit Theory and its Application (1980), Academic Press: Academic Press New York · Zbl 0462.60045
[7] Hanson, D. L.; Russo, P., Some results on increments of the Wiener processes with applications to lag sums of iidrv, Ann. Probab., 11, 609-623 (1983) · Zbl 0519.60030
[8] Heyde, C. C.; Scott, D. J., Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments, Ann. Probab., 1, 428-436 (1973) · Zbl 0259.60021
[9] Ibragimov, I. A., Some limit theorems for stationary process, Theory Probab. Appl., 7, 349-382 (1962) · Zbl 0119.14204
[10] Kolmogorov, A. N.; Rozanov, Y. A., On strong mixing conditions for stationary Gaussian processes, Theory Probab. Appl., 5, 204-208 (1960) · Zbl 0106.12005
[11] Komlós, J.; Major, P.; Tusnády, G., An approximation of partial sums of independent R.V.’s and the sample D.F. I, Z. Wahrsch. Verw. Gebiete, 32, 111-131 (1975) · Zbl 0308.60029
[12] Komlós, J.; Major, P.; Tusnády, G., An approximation of partial sums of independent R.V.’s and the sample D.F. II, Z. Wahrsch. Verw. Gebiete, 34, 35-58 (1976) · Zbl 0307.60045
[13] Moricz, F., A general moment inequality for the maximum of partial sums of singal series, Acta Sci. Math., 44, 67-75 (1982) · Zbl 0487.60025
[14] Peligrad, M., Invariance principles for mixing sequences of random variables, Ann. Probab., 10, 968-981 (1982) · Zbl 0503.60044
[15] Peligrad, M., The \(r\)-quick version of the strong law for stationary ø-mixing sequences, (Edgar, G. A.; Sucheston, L., Almost Everywhere Convergence (1989), Academic Press: Academic Press New York), 335-348
[16] Philipp, W., Invariance principles for independent and weakly dependent random variables, (Eberlein, E.; Taqqu, M. S., Dependence in Probab. and Statist., Progress in Probab. and Statist., 11 (1986)), 225-268
[17] Philipp, W.; Stout, W. F., Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc., 161 (1975) · Zbl 0361.60007
[18] Sakhanenko, A. I., On unimprovable estimates of the rate of convergence in variance principle, (Colloq. Math. Soc. János Bolyai, 32 (1980), Nonparametric Statistical Inference: Nonparametric Statistical Inference Budapest, Hungary), 779-783
[19] Shao, Q.-M., An almost sure invariance principle for Gaussian sequence, Chin. Appl. Probab. Statist., 1, 43-46 (1985), [In Chinese] · Zbl 0571.60046
[20] Shao, Q.-M., Strong approximations on lacunary trigonometric series with weights, Sci. Sinica (A), 30, 796-806 (1987) · Zbl 0654.42015
[21] Shao, Q.-M., A moment inequality and its applications, Acta Math. Sinica, 31, 736-747 (1988), [In Chinese.] · Zbl 0698.60025
[22] Shao, Q.-M., On the complete convergence for ρ-mixing sequence, Acta Math. Sinica, 32, 377-393 (1989), [In Chinese.] · Zbl 0686.60025
[23] Shao, Q.-M., On the invariance principle for ρ-mixing sequences of random variables, Chin. Ann. Math. (B), 10, 427-433 (1989) · Zbl 0683.60023
[24] Shao, Q.-M., Limit theorems for sums of dependent and independent random variables, Ph.D thesis (1989), Univ. of Sci. and Technol. of China: Univ. of Sci. and Technol. of China Anhui, People’s Republic of China, [In Chinese.]
[25] Shao, Q.-M., Strong approximations for independent random variables and their applications (1991), manuscript
[26] Shao, Q.-M.; Lu, C. R., Strong approximations for partial sums of weakly dependent random variables, Sci. Sinica (A), 30, 575-587 (1987) · Zbl 0625.60032
[27] Strassen, V. A., An almost sure invariance principle for the law of the iterated logarithm, Z. Wahrsch. Verw. Gebiete, 3, 211-226 (1964) · Zbl 0132.12903
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