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References

  • Alexander, K. (1984). Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. Prob., 12, 1041–1067.

    Article  MATH  Google Scholar 

  • Alexander, K. (1987). The central limit theorem for empirical processes on Vapnik-Chervonenkis classes, Ann. Prob., 15, 178–203.

    Article  MATH  Google Scholar 

  • Andrews, D. and Pollard, D. (1994). An introduction to functional central limit theorems for dependent stochastic processes, Int. Stat. Rev., 62, 119–132.

    MATH  Google Scholar 

  • Arcones, M. and Yu, B. (1994). Central limit theorems for empirical and U-processes of stationary mixing sequences, J. Theor. Prob., 1, 47–71.

    Article  MathSciNet  Google Scholar 

  • Athreya, K. and Pantula, S. (1986). Mixing properties of Harris chains and autoregressive processes, J. Appl. Prob., 23, 880–892.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran, R. and Millar, P. (1986). Confidence sets for a multinomial distribution, Ann. Stat., 14, 431–443.

    Article  MATH  MathSciNet  Google Scholar 

  • Billingsley, P. (1956). The invariance principle for dependent random variables, Trans. Am. Math. Soc., 83(1), 250–268.

    MATH  MathSciNet  Google Scholar 

  • Billingsley, P. (1968). Convergence of Probability Measures, John Wiley, New York.

    MATH  Google Scholar 

  • Birnbaum, Z. and Marshall, A. (1961). Some multivariate Chebyshev inequalities with extensions to continuous parameter processes, Ann. Math. Stat., 32, 687–703.

    Article  MathSciNet  MATH  Google Scholar 

  • Bradley, R. (2005). Basic properties of strong mixing conditions: a survey and some open problems, Prob. Surv., 2, 107–144.

    Article  MathSciNet  Google Scholar 

  • Brillinger, D. (1969). An asymptotic representation of the sample df, Bull. Am. Math. Soc., 75, 545–547.

    MATH  MathSciNet  Google Scholar 

  • Brown, B. (1971). Martingale central limit theorems, Ann. Math. Stat., 42, 59–66.

    Article  MATH  Google Scholar 

  • Cameron, R. and Martin, W. (1945). Evaluation of various Wiener integrals by use of certain Sturm-Liouville differential equations, Bull. Am. Math. Soc., 51, 73–90.

    MATH  MathSciNet  Google Scholar 

  • Chanda, K. (1974). Strong mixing properties of linear stochastic processes, J. Appl. Prob., 11, 401–408.

    Article  MathSciNet  MATH  Google Scholar 

  • Csörgo, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics, Academic Press, New York.

    Google Scholar 

  • Csörgo, M. and Horváth, L. (1993). Weighted Approximations in Probability and Statistics, John Wiley, New York.

    Google Scholar 

  • Csörgo, M. (1984). Invariance principle for empirical processes, in Handbook of Statistics, P. K. Sen and P. R. Krishraiah (eds.), Vol. 4, North-Holland, Amsterdam, 431–462.

    Google Scholar 

  • Csörgo, M. (2002). A glimpse of the impact of Paul Erdös on probability and statistics, Can. J. Stat., 30(4), 493–556.

    Google Scholar 

  • Csörgo, S. and Hall, P. (1984). The KMT approximations and their applications, Aust. J. Stat., 26(2), 189–218.

    Google Scholar 

  • Donsker, M. (1951). An invariance principle for certain probability limit theorems, Mem. Am. Math. Soc., 6.

    Google Scholar 

  • Doukhan, P. (1994). Mixing: Properties and Examples, Lecture Notes in Statistics, Vol. 85, Springer, New York.

    Google Scholar 

  • Dudley, R. (1978). Central limit theorems for empirical measures, Ann. Prob., 6, 899–929.

    Article  MATH  MathSciNet  Google Scholar 

  • Dudley, R. (1979). Central limit theorems for empirical measures, Ann. Prob., 7(5), 909–911.

    Article  MATH  MathSciNet  Google Scholar 

  • Dudley, R. and Philipp, W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahr. Verw. Geb., 62, 509–552.

    Article  MATH  MathSciNet  Google Scholar 

  • Dudley, R. (1984). A Course on Empirical Processes, Lecture Notes in Mathematics, Springer, Berlin.

    Google Scholar 

  • Durrett, R. (1996). Probability: Theory and Examples, 2nd ed., Duxbury Press, Belmont, CA.

    Google Scholar 

  • Einmahl, U. (1987). Strong invariance principles for partial sums of independent random vectors, Ann. Prob., 15(4), 1419–1440.

    Article  MATH  MathSciNet  Google Scholar 

  • Erdös, P. and Kac, M. (1946). On certain limit theorems of the theory of Probability, Bull. Am. Math. Soc., 52, 292–302.

    MATH  Google Scholar 

  • Fitzsimmons, P. and Pitman, J. (1999). Kac’s moment formula and the Feynman-Kac formula for additive functionals of a Markov process, Stoch. Proc. Appl., 79(1), 117–134.

    Article  MATH  MathSciNet  Google Scholar 

  • Gastwirth, J. and Rubin, H. (1975). The asymptotic distribution theory of the empiric cdf for mixing stochastic processes, Ann. Stat., 3, 809–824.

    Article  MATH  MathSciNet  Google Scholar 

  • Giné, E. (1996). Empirical processes and applications: an overview, Bernoulli, 2(1), 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  • Giné, E. and Zinn, J. (1984). Some limit theorems for empirical processes, with discussion, Ann. Prob., 12(4), 929–998.

    Article  MATH  Google Scholar 

  • Hall, P. (1977). Martingale invariance principles, Ann. Prob., 5(6), 875–887.

    Article  MATH  Google Scholar 

  • Hall, P. and Heyde, C. (1980). Martingale Limit Theory and Its Applications, Academic Press, New York.

    Google Scholar 

  • Heyde, C. (1981). Invariance principles in statistics, Int. Stat. Rev., 49(2), 143–152.

    MATH  MathSciNet  Google Scholar 

  • Jain, N., Jogdeo, K., and Stout, W. (1975). Upper and lower functions for martingales and mixing processes, Ann. Prob., 3, 119–145.

    Article  MATH  MathSciNet  Google Scholar 

  • Kac, M. (1951). On some connections between probability theory and differential and integral equations, in Proceedings of the Second Berkeley Symposium, J. Neyman (ed.), University of California Press, Berkeley, 189–215.

    Google Scholar 

  • Kiefer, J. (1972). Skorohod embedding of multivariate rvs and the sample df, Z. Wahr. Verw. Geb., 24, 1–35.

    Article  MATH  Google Scholar 

  • Kolmogorov, A. (1933). Izv. Akad. Nauk SSSR, 7, 363–372 (in German).

    Google Scholar 

  • Komlós, J., Major, P., and Tusnady, G. (1975). An approximation of partial sums of independent rvs and the sample df: I, Z. Wahr. Verw. Geb., 32, 111–131.

    Article  MATH  Google Scholar 

  • Komlós, J., Major, P., and Tusnady, G. (1976). An approximation of partial sums of independent rvs and the sample df: II, Z. Wahr. Verw. Geb., 34, 33–58.

    Article  MATH  Google Scholar 

  • Koul, H. (1977). Behavior of robust estimators in the regression model with dependent errors, Ann. Stat., 5, 681–699.

    Article  MATH  MathSciNet  Google Scholar 

  • Major, P. (1978). On the invariance principle for sums of iid random variables, J. Multivar. Anal., 8, 487–517.

    Article  MATH  MathSciNet  Google Scholar 

  • Mandrekar, V. and Rao, B.V. (1989). On a limit theorem and invariance principle for symmetric statistics, Prob. Math. Stat., 10, 271–276.

    MATH  MathSciNet  Google Scholar 

  • Massart, P. (1989). Strong approximation for multivariate empirical and related processes, via KMT construction, Ann. Prob., 17(1), 266–291.

    Article  MATH  MathSciNet  Google Scholar 

  • McLeish, D. (1974). Dependent central limit theorems and invariance principles, Ann. Prob., 2, 620–628.

    Article  MATH  MathSciNet  Google Scholar 

  • McLeish, D. (1975). Invariance principles for dependent variables, Z. Wahr. Verw. Geb., 3, 165–178.

    Article  Google Scholar 

  • Merlevéde, F., Peligrad, M., and Utev, S. (2006). Recent advances in invariance principles for stationary sequences, Prob. Surv., 3, 1–36.

    Article  Google Scholar 

  • Oblój, J. (2004). The Skorohod embedding problem and its offspring, Prob. Surv., 1, 321–390.

    Article  Google Scholar 

  • Philipp, W. (1979). Almost sure invariance principles for sums of B-valued random variables, in Problems in Banach Spaces, A. Beck (ed.), Vol. II, Lecture Notes in Mathematics, Vol. 709, Springer, Berlin, 171–193.

    Google Scholar 

  • Philipp, W. and Stout, W. (1975). Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Am. Math. Soc., 2, 161.

    Google Scholar 

  • Pollard, D. (1989). Asymptotics via empirical processes, Stat. Sci., 4, 341–366.

    Article  MATH  MathSciNet  Google Scholar 

  • Pyke, R. (1984). Asymptotic results for empirical and partial sum processes: a review, Can. J. Stat., 12, 241–264.

    Article  MATH  MathSciNet  Google Scholar 

  • Révész, P. (1976). On strong approximation of the multidimensional empirical process, Ann. Prob., 4, 729–743.

    Article  MATH  Google Scholar 

  • Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition, Proc. Natl. Acad. Sci. USA, 42, 43–47.

    Article  MATH  MathSciNet  Google Scholar 

  • Rosenblatt, M. (1972). Uniform ergodicity and strong mixing, Z. Wahr. Verw. Geb., 24, 79–84.

    Article  MATH  MathSciNet  Google Scholar 

  • Sauer, N. (1972). On the density of families of sets, J. Comb. Theory Ser. A, 13, 145–147.

    Article  MATH  MathSciNet  Google Scholar 

  • Sen, P.K. (1978). An invariance principle for linear combinations of order statistics, Z. Wahr. Verw. Geb., 42(4), 327–340.

    Article  MATH  Google Scholar 

  • Shorack, G. and Wellner, J. (1986). Empirical Processes with Applications to Statistics, John Wiley, New York.

    MATH  Google Scholar 

  • Smirnov, N. (1944). Approximate laws of distribution of random variables from empirical data, Usp. Mat. Nauk., 10, 179–206.

    MATH  Google Scholar 

  • Strassen, V. (1964). An invariance principle for the law of the iterated logarithm, Z. Wahr. Verw. Geb., 3, 211–226.

    Article  MATH  MathSciNet  Google Scholar 

  • Strassen, V. (1967). Almost sure behavior of sums of independent random variables and martingales, in Proceedings of the Fifth Berkeley Symposium, L. Le Cam and J. Neyman (eds.), Vol. 1 University of California Press, Berkeley, 315–343.

    Google Scholar 

  • van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes, Springer-Verlag, New York.

    MATH  Google Scholar 

  • Vapnik, V. and Chervonenkis, A. (1971). On the uniform convergence of relative frequencies of events to their probabilities, Theory Prob. Appl., 16, 264–280.

    Article  MATH  Google Scholar 

  • Wellner, J. (1992). Empirical processes in action: a review, Int. Stat. Rev., 60(3), 247–269.

    MATH  Google Scholar 

  • Whithers, C. (1981). Conditions for linear processes to be strong mixing, Z. Wahr. Verw. Geb., 57, 477–480.

    Article  Google Scholar 

  • Whitt, W. (1980). Some useful functions for functional limit theorems, Math. Oper. Res., 5, 67–85.

    MATH  MathSciNet  Google Scholar 

  • Yu, B. (1994). Rates of convergence for empirical processes of stationary mixing sequences, Ann. Prob., 22, 94–116.

    Article  MATH  Google Scholar 

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DasGupta, A. (2008). Invariance Principles. In: Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75971-5_12

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