A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences. (English) Zbl 0792.60018
With the help of an extension of Hoeffding’s equality, we develop a way for estimating the covariance structures for empirical functions of associated sequences in terms of covariances of the original random variables. Based on these estimations, a Glivenko-Cantelli lemma for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained, all under the conditions on covariances of the original random variables.
Reviewer: H.Yu (Ottawa)
Keywords:
notions of positive dependence; Hoeffding’s equality; Glivenko-Cantelli lemma; weak convergence for empirical processesReferences:
| [1] | Barlow, R.E., Proschan, F.: Statistical theory of reliability and life testing: Probability models. Silver Spring, MD: to begin with publishes 1981 · Zbl 0379.62080 |
| [2] | Billingsley, P.: Convergence of probability measures New York: Wiley 1968 · Zbl 0172.21201 |
| [3] | Birkel, T.: The invariance principle for associated processes. Stochastic Processes Appl.27, 57-71 (1987) · Zbl 0632.60001 · doi:10.1016/0304-4149(87)90005-6 |
| [4] | Birkel, T.: A note on the strong law of large numbers for positively dependent random variables. stat. Probab. Lett.7, 17-20 (1989) · Zbl 0661.60048 · doi:10.1016/0167-7152(88)90080-6 |
| [5] | Block, H.W., Fang, Z.: A multivariate extension of Höeffding’s lemma. Ann. Probab.16, 1803-1820 (1988) · Zbl 0651.62042 · doi:10.1214/aop/1176991598 |
| [6] | Burton, R.M., Dabrowski, A.R., Dehling, H.: An invariance principle for weakly associated random variables. Stochastic Processes Appl.23, 301-306 (1986) · Zbl 0611.60028 · doi:10.1016/0304-4149(86)90043-8 |
| [7] | Chung, K.L.: A course in probability theory, 2nd edn. New York: Academic Press 1974 · Zbl 0345.60003 |
| [8] | Cox, J.T., Grimmett, G.: Central limit theorems for associated random variables and the percolation model. Ann. Probab.12, 514-528 (1984) · Zbl 0536.60094 · doi:10.1214/aop/1176993303 |
| [9] | Csörg?, M., Csörg?, S., Horváth, L.: An asymptotic theory for empirical reliability and concentration processes. (Lect. Notes Stat., vol. 33) Berlin Heidelberg New York: Springer 1986 · Zbl 0605.62105 |
| [10] | Dabrowski, A.R., Dehling, H.: A Berry-Esséen theorem and a functional law of the iterated logarithm for weakly associated random variables. Stochastic Processes Appl.30, 277-289 (1988) · Zbl 0665.60027 · doi:10.1016/0304-4149(88)90089-0 |
| [11] | Esary, J., Proschan, F., Walkup, D.: Association of random variables with applications. Ann. Math. Stat.38, 1466-1474 (1967) · Zbl 0183.21502 · doi:10.1214/aoms/1177698701 |
| [12] | Höeffding, W.: Masstabinvariante Korrelations-theorie. Schr. Math. Inst. Univ. Berlin.5, 181-233 (1940) |
| [13] | Joag-Dev, K.: Independence via uncorrelatedness under certain dependence structures. Ann. Probab.11, 1037-1041 (1983) · Zbl 0542.62046 · doi:10.1214/aop/1176993452 |
| [14] | Jogdeo, K.: Characterizations of independence in certain families of bivariate and multivariate distributions. Ann. Math. Stat.39, 433-441 (1968) · Zbl 0164.49106 · doi:10.1214/aoms/1177698407 |
| [15] | Lebowitz, J.: Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems. Commun. Math. Phys.28, 313-321 (1972) · doi:10.1007/BF01645632 |
| [16] | Newman, C.M.: Normal fluctuations and the FKG inequalities. Commun. Math. Phys.74, 119-128 (1980) · Zbl 0429.60096 · doi:10.1007/BF01197754 |
| [17] | Newman, C.M.: A general central limit theorem for FKG systems. Commun. Math. Phys.91, 75-80 (1983) · Zbl 0528.60024 · doi:10.1007/BF01206051 |
| [18] | Newman, C.M.: Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y.L. (ed.) Inequalities in statistics and probability. Hayward, CA: Inst. Math. Stat. 1984 |
| [19] | Newman, C.M., Wright, A.L.: An invariance principle for certain dependent sequences. Ann. Probab.9, 671-675 (1981) · Zbl 0465.60009 · doi:10.1214/aop/1176994374 |
| [20] | Wood, T.E.: A Berry-Esséen theorem for associated random variables. Ann. Probab.11, 1042-1047 (1983) · Zbl 0522.60017 · doi:10.1214/aop/1176993453 |
| [21] | Wood, T.E.: A local limit theorem for associated sequences. Ann. Probab.13, 625-629 (1985) · Zbl 0563.60028 · doi:10.1214/aop/1176993015 |
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