Abstract
By combining the Csörgő–Révész quantile transform methods and the Skorohod–Strassen martingale embedding theorem, we prove a strong approximation theorem for quasi-associated random variables with mean zero and finite (2+δ)th moment under polynomial decay rate. As a consequence, the decay rate for a strong approximation theorem of associated sequences of Yu (1996) is weakened.
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Esary, J., Proschan, F., Walkup, D.: Association of random variables with applications. Ann. Math. Statist., 38, 1466–1474 (1967)
Newman, C. M.: Normal fluctuations and the FKG inequalities. Comm. Math. Phys., 74, 119–128 (1980)
Cox, J. T., Grimmett, G.: Central limit theorems for associated random variables and the percolation model. Ann. Probab., 12, 514–528 (1984)
Burton, R. M., Dabrowski, A. R., Dehling, H.: An invariance principle for weakly associated random variables. Stoch. Process. Appl., 23, 301–306 (1986)
Dabrowski, A. R., Dehling, H.: A Berry–Esseen theorem and a functional law of the iterated logarithm for weakly associated random variables. Stoch. Process. Appl., 30, 277–289 (1988)
Khoshnevisan, D., Lewis, T. M.: A law of the iterated logarithm for stable processes in random scenery. Stoch. Process. Appl., 74, 89–121 (1998)
Yu, H.: A strong invariance principle for associated sequences. Ann. Probab., 24, 2079–2097 (1996)
Berkes, I., Philipp, W.: Approximation theorems for independent and weakly dependence random variables. Ann. Probab., 7, 29–54 (1979)
Csörgő, M., Révész, P.: A new method to prove Strassen type laws of invariance principle I. Z. Wahrsch. Verw. Gebiete, 31, 255–260 (1975)
Csörgő, M., Révész, P.: A new method to prove Strassen type laws of invariance principle II. Z. Wahrsch. Verw. Gebiete, 31, 261–269 (1975)
Lewis, T. M.: Limit theorems for partial sums of quasi–associated random variables, Asymptotic Methods in Probability and Statistics, B. Szyszkowicz (Editor), 31–48, 1998
Birkel, T.: Moment bounds for associated sequences. Ann. Probab., 16, 1184–1193 (1988)
Newman, C. M., Wright, A. L.: An invariance principle for certain dependent sequences. Ann. Probab., 9, 671–675 (1981)
Price, G. B.: Bounds for determinants with dominant principle diagonal. Proc. Amer. Math. Soc., 2, 497–502 (1951)
Hall, P., Heyde, C. C.: Martingale Limit Theory and its Application, Academic Press, New York, 1980
Chow, Y. S.: Local convergence of martingale and the law of large numbers. Ann. Math. Statist., 36, 552–558 (1965)
Hanson, D. L., Russo, Ralph. P.: Some results on increments of the Wiener processes with applications to lag sums of iidrv. Ann. Probab., 11, 609–623 (1983)
Csörgő, M., Révész, P.: Strong Approximations in Probability and Statistics, Academic Press, New York, 1981
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Research supported by NSFC (10401037) and China Postdoctoral Science Foundation
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Wang, W.S. A Strong Approximation Theorem for Quasi-associated Sequences. Acta Math Sinica 21, 1269–1276 (2005). https://doi.org/10.1007/s10114-004-0471-7
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DOI: https://doi.org/10.1007/s10114-004-0471-7