Abstract
The purpose of this paper is to describe the progress that has recently been made in the study of the central limit theorem and its weak invariance principle for mixing sequences of random variables and to point out some open problems in this subject.
The work was support in part by a NSF grant DMS-8503016 and by a Taft Grant in aid for travel from the Charles Phelps Taft Memorial Fund, University of Cincinnati.
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Bibliography
Berbee, H. C. P., (1979); Random Walks With Stationary Increments and Renewal Theory. Mathematical Centre, Amsterdam.
Boldrighini, C., De Massi, A., Nogueira, A., Presutti, E., (1985); The Dynamics of a particle interacting with a semi-infinite ideal gas in a Bernoulli flow. Statistical Physics and Dynamical Systems (Progress in Physics) Birkhäuser, p. 153.
Bradley, R. C. (1983), Equivalent Measures of Dependence, J. Multivariate Analysis 13, 167–176.
Bradley, R. C. (1980), A Remark on the Central Limit Question for Dependent Random Variances, J. Appl. Prob. 17, 94–101.
Bradley, R. C. (1981 ), A Sufficient Condition for Linear Growth of Variables in a Stationary Random Sequence, Proc. Amer. Math. Soc. 83, 586–589.
Bradley, R. C. (1981 ), Central Limit Theorems under Weak Dependence, J. Multivariate Analysis, 11, 1–16.
Bradley, R. C. (1984 ), On the Central Limit Question under Absolute Regularity (to appear in Ann. of Prob.).
Bradley, R. C. ( 1984 ), The Central Limit Question under a-mixing (to appear in Rocky Mtn. J. Math.).
Bradley, R. C. (1983 ), Information Regularity and Central Limit Question, Rocky Mtn. J. Math., 13, 77–97.
Bradley, R. C., Bryc, W. (1985), Multilinear Forms and Measures of Dependence Between Random Variables, J. Multivariate Analysis, 16, 335–367.
Bradley, R. C. (1982), Counterexamples to the Central Limit Theorem Under Strong Mixing Conditions, Colloquia Mathematica Soc. Jénos Bolyai. 36 Limit Theorems in Probability and Statistics, Veszprém (Hungary).
Bradley, R. C. (1985), The Basic Properties of Strong Mixing Conditions, (to appear in Dependence in Probability and Statistics, Birkhauser).
Bradley, R. C., Peligrad, M. (1985), Invariance Principles Under A Two-Part Mixing Assumption. Preprint.
Bryc, W. (1985), Central Limit Theorem for 4-mixing Random Sequences Under Lindeberg-type Condition, (preprint).
Cogburn, R. (1960), Asymptotic Properties of Stationary Sequences, Univ. Calif. Publ. Statist. 3, 99–146.
Davydov, Y. A. (1968), Convergence of Distributions Generated by Stationary Stochastic Processes., Teor. Probability Appl. 13, 691–696.
Davydov, Y. A. (1969), On the Strong Mixing Property for Markov Chains with a Countable Number of States, Soviet Math. Doki. 10, 825–827.
Davydov, Y. A. (1973), Mixing Conditions for Markov Chains, Theory Probab. Appl. 18, 312–328.
Dehling, H., Denker, M., Philipp, W., (1984), Central Limit Theorems for Mixing Sequences of Random Variables under Minimal Conditions (to appear in Ann. Probability).
Dehling, H. and Philipp, W. (1982), Almost Sure Invariance Principle for Weakly Dependent Vector-Valued Random Variables, Ann. Prob. 10, 689–701.
Denker, M. (1978), A Note on the Central Limit Theorem for Strongly Mixing Random Variables. Preprint.
Denker, M. (1979), A Limit Theorem for Mixing Stationary Processes and its Applications. Preprint.
Denker, M. (1985), Uniform Integrability and the Central Limit Theorem (to appear in Dependence in Probability and Statistics. Birkhäuser).
Dvoretzky, A. (1972), Asymptotic Normality for Sums of Dependent Random Variables, Proc. Sixth Berkeley Sympos. Math. Stat. Prob., Vol. 2, 515–535.
Eberlein, E. (1984), On Strong Invariance Principle Under Dependence Assumptions, (to appear in Ann. Probability).
Gastwirth, J. L., Rubin, H. (1975), The Asymptotic Distribution Theory of the Empiric c.d.f. for Mixing Stochastic Processes, Ann. Statist., 3, 809–824.
Hall, P., Heyde, C. C. (1980), Martingale Limit Theory and Its Applications, Academic Press, New York.
Hahn, M. G., Kuelbs, J. (1985), Asymptotic Normality of Trimmed Sums of + Mixing Random Variables. Preprint.
Herrndorf, N. (1983 ), The Invariance Principle for +-mixing Sequences, Z. Wahrsch. Verw. Gebiete 63, 1, 97–109.
Herrndorf, N., (1983 ), Stationary Strongly Mixing Sequences Not Satisfying the Central Limit Theorem, Annals Prob. 11, 809–813.
Herrndorf, N., ( 1983 ), A Functional Central Limit Theorem for Strongly Mixing Sequences of Random Variables, (Z. Wahrsch. verw. Gebiete, submitted).
Herrndorf, N., (1984 ), A Functional Central Limit Theorem for P-Mixing Sequences, J. of Multiv. Anal. 15, 141–146.
Herrndorf, N., (1984 ), A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables, Ann. of Prob., 12, 141–153.
Hoffman-Jorgensen, J., (1974), Sums of Independent Banach Space Valued Random Variables, Studia Math. 52, 159–186.
Ibragimov, I. A., (1962), Some Limit Theorem for Stationary Processes, Theory Prob. Appl. 7, 349–382.
Ibragimov, I. A., (1975), A Note on the Central Limit Theorem for Dependent Random Variables, Theory Prob. Appl. 20, 135–141.
Ibragimov, I. A., Linnik, Y. V. (1971), Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
Ibragimov, I. A., Rozanov, Y. A., (1978), Gaussian Random Processes, Springer-Verlag, Berlin.
Iosifescu, M. (1980), Recent Advances in Mixing Sequences of Random Variables. Third International Summer School on Probability Theory and Mathematical Statistics, Varna. 1978.
Iosifescu, M., Teodorescu, R. (1969), Random Processes and Learning, Springer-Verlag, New York.
Jakubowski, A. (1985), A Note on the Invariance Principle for Stationary 4-Mixing Sequences. Tightness via Stopping Times. Preprint.
Kesten, H., O’Brien, G. L. (1976), Examples of Mixing Sequences, Duke Math. J. 43, 405–415.
Kolmogorov, A. N., Rozanov, Y. A. (1960), On Strong Mixing Conditions for Stationary Gaussian Processes. Theory Probab. Appl. 5, 204–208.
Lai, T. L. (1977), Convergence Rates and r-Quick Version of the Strong Law for Stationary Mixing Sequences, Ann. Prob. 5, 693–706.
Lai, T. L., Robbins, H. (1978), A Class of Dependent Random Variables and Their Maxima, Z. Wahrsch. Verw. Gebiete 42, 89–111.
McLeish, D. L. (1975), Invariance Principles for Dependent Variables, Z. Wahrsch. verw. Gebiete 32, 165–178.
McLeish, D. L. (1977), On the Invariance Principles for Nonstationary Mixingales, Ann. Prob. 5, 616–621.
Moricz, F. (1982), A General Moment Inequality for the Maximum of Partial Sums of Single Series, Acta Sci. Math. 44, 67–75.
Moricz, F., Serfling, R. J. and Stout, W. (1982), Moment and Probability Bounds with Quasi-Superadditive Structure for the Maximum Partial Sum, Ann. Prob. 10, 1032–1040.
Nahapetian, B. S. (1984), On Limit Theorem for Dependent Random Variables. The Sixth International Symposium on Information Theory. September 18–22, 1984, Tashkent.
Oodaira, H. and Yoshihara, K. (1971), The Law of the iterated Logarithm for Stationary Processes Satisfying Mixing Conditions, Kodai Math. Sem. Rep. 23, 311–334.
Oodaira, H. and Yoshihara, K. (1972), Functional Central Limit Theorems for Strictly Stationary Processes Satisfying the Strong Mixing Condition, Kodai Math. Sem. Rep. 24, 259–269.
Peligrad, M. (1981 ), An Invariance Principle for Dependent Random Variables, Z. Wahrsch. verw. Gebiete 57, 495–507.
Peligrad, M. ( 1981 ), A Maximal Inequality for 4-Mixing Sequences, Center of Math. Statistics Notes, Bucharest.
Peligrad, M. (1982), Invariance Principles for Mixing Sequences of Random Variables, The Ann. of Prob. 10, 4, 968–981.
Peligrad, M. (1983), A Note on Two Measures of Dependence and Mixing Sequences, Adv. Appl. Prob. 15, 461–464.
Peligrad, M. (1984 ), An Invariance Principle for 4-Mixing Sequences, (to appear in Ann. of Prob.).
Peligrad, M. (1984 ), Invariance Principles under Weak Dependence (to appear in J. of Multivariate Analysis).
Philipp, W. (1979), An Almost Sure Invariance Principle for Sums of B-Valued Random Variables, Lecture Notes In Math. 709, 171–193, Springer-Verlag, New York.
Philipp, W. (1980), Weak and Lu-Invariance Principle for Sums of B-Valued Random Variables, The Ann. of Prob. 8, 1, 68–82.
Philipp, W., Stout, W. F. (1975), Almost Sure Invariance Principles for Sums of Weakly Dependent Random Variables, Amer. Math. Soc. Memoir, No. 161.
Philipp, W., Webb, G. R. (1973), An Invariance Principle for Mixing Sequences of Random Variables, Z. Warhsch. verw. Gebiete 25, 223–237.
Rosenblatt, M. (1956), A Central Limit Theorem and a Strong Mixing Condition, Proc. Nat. Acad. Sci. USA 42, 43–47.
Rosenblatt, M. (1971), Markov Processes, Structure and Asymptotic Behavior, Springer-Verlag, Berlin.
SamAr, J. D 19x4), ectorsgeitn stationarry Rows, Ann. robar12, rrays of an om Vectors w 390–426.
Serfling, R. J. (1968), Contributions to Central Limit Theory for Dependent Variables, Ann. Math. Stat. 39, 1158–1175.
Shields, P., (1973), The Theory of Bernoulli Shifts. University of Chicago Press, Chicago.
Simons, G., Stout, W. (1978), A Weak Invariance Principle with Applications to Domains of Attraction, Ann. Prob. 6, 294–315.
Volkonskii, V. A., Rozanov, Y. A. (1959), Some Limit Theorems for Random Functions I. Theory Prob. Appl. 4, 178–197.
Yokoyama, R. (1980), Moment Bounds for Stationary Mixing Sequences, Z. Wahrsch. verw. Gebiete 52, 45–57.
Yoshihara, K. (1978), Probability Inequalities for Sums of Absolutely Regular Processes and Their Applications, Z. Wahrsch. verw. Gebiete 43, 319–329.
Withers, C. L. (1981), Central Limit Theorems for Dependent Variables I, Z. Wahrsch. verw. Gebiete 57, 509–534.
Withers, C. L. (1983), Corringendum to Central Limit Theorems for Dependent Random Variables, Z. Wahrsch. verw. Gebiete, 63, 555.
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Peligrad, M. (1986). Recent Advances in the Central Limit Theorem and Its Weak Invariance Principle for Mixing Sequences of Random Variables (A Survey). In: Eberlein, E., Taqqu, M.S. (eds) Dependence in Probability and Statistics. Progress in Probability and Statistics, vol 11. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4615-8162-8_9
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