## A strong invariance principle for associated sequences.(English)Zbl 0879.60028

Summary: By combining the Berkes-Philipp blocking technique and the Csörgö-Révész quantile transform methods, we find that partial sums of an associated sequence can be approximated almost surely by partial sums of another sequence with Gaussian marginals. A crucial fact is that this latter sequence is still associated with covariances roughly bounded by the covariances of the original sequence, and that one can approximate it by an i.i.d. Gaussian process using the Berkes-Philipp method. We require that the original sequence has finite $$(2+r)$$th moments, $$r>0$$, and a power decay rate of a coefficient $$u(n)$$ which describes the covariance structure of the sequence. Based on this result, we obtain a strong invariance principle for associated sequences if $$u(n)$$ exponentially decreases to 0.

### MSC:

 60F17 Functional limit theorems; invariance principles 60B10 Convergence of probability measures 60F15 Strong limit theorems 62G30 Order statistics; empirical distribution functions
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### References:

 [1] BARLOW, R. E. and PROSCHAN, F. 1981. Statistical Theory of Reliability and Life Testing: Probability Models. To Begin with Publisher, Silver Spring, MD. Z. · Zbl 0379.62080 [2] BERBEE, H. 1987. Convergence rates in the strong law for bounded mixing sequences. Probab. Theory Related Fields 74 255 270. Z. · Zbl 0587.60028 [3] BERKES, I. and PHILIPP, W. 1979. Approximation theorems for independent and weakly dependent random variables. Ann. Probab. 7 29 54. Z. · Zbl 0392.60024 [4] BILLINGSLEY, P. 1968. Convergence of Probability Measures. Wiley, New York. Z. · Zbl 0172.21201 [5] BIRKEL, T. 1987. The invariance principle for associated processes. Stochastic Process. Appl. 27 57 71. Z. · Zbl 0632.60001 [6] BIRKEL, T. 1988a. On the convergence rate in the central limit theorems for associated processes. Ann. Probab. 16 1685 1698. Z. · Zbl 0658.60039 [7] BIRKEL, T. 1988b. Moment bounds for associated sequences. Ann. Probab. 16 1184 1193. Z. · Zbl 0647.60039 [8] BURTON, R. M., DABROWSKI, A. R. and DEHLING, H. 1986. An invariance principle for weakly associated random variables. Stochastic Process. Appl. 23 301 306. Z. · Zbl 0611.60028 [9] COX, J. T. and GRIMMETT, G. 1984. Central limit theorems for associated random variables and the percolation model. Ann. Probab. 12 514 528. Z. · Zbl 0536.60094 [10] CSORGO, M. and REVESZ, P. 1975a. A new method to prove Strassen ty pe laws of invariance \" \' ṕrinciple. I. Z. Wahrsch. Verw. Gebiete 31 255 260. Z. [11] CSORGO, M. and REVESZ, P. 1975b. A new method to prove Strassen ty pe laws of invariance \" \' ṕrinciple. II. Z. Wahrsch. Verw. Gebiete 31 261 269. Z. [12] CSORGO, M. and REVESZ, P. 1981. Strong Approximations in Probability and Statistics. Aca\" \' \' demic Press, New York. Z. [13] DABROWSKI, A. R. and DEHLING, H. 1988. A Berry Esseen theorem and a functional law of the iterated logarithm for weakly associated random variables. Stochastic Process. Appl. 30 277 289. Z. · Zbl 0665.60027 [14] ESARY, J., PROSCHAN, F. and WALKUP, D. 1967. Association of random variables with applications. Ann. Math. Statist. 38 1466 1474. Z. · Zbl 0183.21502 [15] FELLER, W. 1971. An Introduction to Probability Theory and its Applications 2, 2nd ed. John Wiley, New York. Z. · Zbl 0219.60003 [16] KOMLOS, J., MAJOR, P. and TUSNADY, G. 1975. An approximation of partial sums of independent Ŕ.V.’s and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111 131. Z. [17] KOMLOS, J., MAJOR, P. and TUSNADY, G. 1976. An approximation of partial sums of independent Ŕ.V.’s and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 35 58. Z. [18] LOEVE, M. 1977. Probability Theory 1, 4th ed. Springer, New York. Ź. · Zbl 0359.60001 [19] NEWMAN, C. M. 1980. Normal fluctuations and the FKG inequalities. Comm. Math. Phy s. 74 119 128. · Zbl 0429.60096 [20] NEWMAN, C. M. 1983. A general central limit theorem for FKG sy stems. Comm. Math. Phy s. 91 75 80. Z. · Zbl 0528.60024 [21] NEWMAN, C. M. and WRIGHT, A. L. 1981. An invariance principle for certain dependent sequences. Ann. Probab. 9 671 675. Z. · Zbl 0465.60009 [22] PHILIPP, W. 1986. Invariance principles for independent and weakly dependent random variZ. ables. In Dependence in Probability and Statistics E. Eberlein and M. Taqque, eds. 225 268. Birkhauser, Boston. \" Z. [23] PHILIPP, W. and STOUT, W. F. 1975. Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 161. Z. · Zbl 0361.60007 [24] PITT, L. D. 1982. Positively correlated normal variables are associated. Ann. Probab. 10 496 499. Z. · Zbl 0482.62046 [25] WOOD, T. E. 1983. A Berry Esseen theorem for associated random variables. Ann. Probab. 11 1042 1047. Z. · Zbl 0522.60017 [26] WOOD, T. E. 1985. A local limit theorem for associated sequences. Ann. Probab. 13 625 629. Z. · Zbl 0563.60028 [27] YU, H. 1985. An invariance principle for associated sequences of random variables. J. Engrg. Math. 2 55 60. Z. [28] YU, H. 1986. The law of the iterated logarithm for associated random variables. Acta Math. Sinica 29 507 511. Z. · Zbl 0615.60031 [29] YU, H. 1993. A Glivenko Cantelli lemma and weak convergence for empirical processes of associated sequences. Probab. Theory Related Fields 95 357 370. · Zbl 0792.60018 [30] LONDON, ONTARIO N6A 5B7 CANADA E-MAIL: hy u@fisher.stats.uwo.ca
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