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Invariance Principles

  • Anirban DasGupta
Part of the Springer Texts in Statistics book series (STS)

Keywords

Central Limit Theorem Gaussian Process Invariance Principle Empirical Process Brownian Bridge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Alexander, K. (1984). Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. Prob., 12, 1041–1067.MATHCrossRefGoogle Scholar
  2. Alexander, K. (1987). The central limit theorem for empirical processes on Vapnik-Chervonenkis classes, Ann. Prob., 15, 178–203.MATHCrossRefGoogle Scholar
  3. Andrews, D. and Pollard, D. (1994). An introduction to functional central limit theorems for dependent stochastic processes, Int. Stat. Rev., 62, 119–132.MATHGoogle Scholar
  4. Arcones, M. and Yu, B. (1994). Central limit theorems for empirical and U-processes of stationary mixing sequences, J. Theor. Prob., 1, 47–71.CrossRefMathSciNetGoogle Scholar
  5. Athreya, K. and Pantula, S. (1986). Mixing properties of Harris chains and autoregressive processes, J. Appl. Prob., 23, 880–892.MATHCrossRefMathSciNetGoogle Scholar
  6. Beran, R. and Millar, P. (1986). Confidence sets for a multinomial distribution, Ann. Stat., 14, 431–443.MATHCrossRefMathSciNetGoogle Scholar
  7. Billingsley, P. (1956). The invariance principle for dependent random variables, Trans. Am. Math. Soc., 83(1), 250–268.MATHMathSciNetGoogle Scholar
  8. Billingsley, P. (1968). Convergence of Probability Measures, John Wiley, New York.MATHGoogle Scholar
  9. Birnbaum, Z. and Marshall, A. (1961). Some multivariate Chebyshev inequalities with extensions to continuous parameter processes, Ann. Math. Stat., 32, 687–703.CrossRefMathSciNetMATHGoogle Scholar
  10. Bradley, R. (2005). Basic properties of strong mixing conditions: a survey and some open problems, Prob. Surv., 2, 107–144.CrossRefMathSciNetGoogle Scholar
  11. Brillinger, D. (1969). An asymptotic representation of the sample df, Bull. Am. Math. Soc., 75, 545–547.MATHMathSciNetGoogle Scholar
  12. Brown, B. (1971). Martingale central limit theorems, Ann. Math. Stat., 42, 59–66.CrossRefMATHGoogle Scholar
  13. 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.MATHMathSciNetGoogle Scholar
  14. Chanda, K. (1974). Strong mixing properties of linear stochastic processes, J. Appl. Prob., 11, 401–408.CrossRefMathSciNetMATHGoogle Scholar
  15. Csörgo, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics, Academic Press, New York.Google Scholar
  16. Csörgo, M. and Horváth, L. (1993). Weighted Approximations in Probability and Statistics, John Wiley, New York.Google Scholar
  17. 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
  18. Csörgo, M. (2002). A glimpse of the impact of Paul Erdös on probability and statistics, Can. J. Stat., 30(4), 493–556.Google Scholar
  19. Csörgo, S. and Hall, P. (1984). The KMT approximations and their applications, Aust. J. Stat., 26(2), 189–218.Google Scholar
  20. Donsker, M. (1951). An invariance principle for certain probability limit theorems, Mem. Am. Math. Soc., 6.Google Scholar
  21. Doukhan, P. (1994). Mixing: Properties and Examples, Lecture Notes in Statistics, Vol. 85, Springer, New York.Google Scholar
  22. Dudley, R. (1978). Central limit theorems for empirical measures, Ann. Prob., 6, 899–929.MATHCrossRefMathSciNetGoogle Scholar
  23. Dudley, R. (1979). Central limit theorems for empirical measures, Ann. Prob., 7(5), 909–911.MATHCrossRefMathSciNetGoogle Scholar
  24. 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.MATHCrossRefMathSciNetGoogle Scholar
  25. Dudley, R. (1984). A Course on Empirical Processes, Lecture Notes in Mathematics, Springer, Berlin.Google Scholar
  26. Durrett, R. (1996). Probability: Theory and Examples, 2nd ed., Duxbury Press, Belmont, CA.Google Scholar
  27. Einmahl, U. (1987). Strong invariance principles for partial sums of independent random vectors, Ann. Prob., 15(4), 1419–1440.MATHCrossRefMathSciNetGoogle Scholar
  28. Erdös, P. and Kac, M. (1946). On certain limit theorems of the theory of Probability, Bull. Am. Math. Soc., 52, 292–302.MATHGoogle Scholar
  29. 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.MATHCrossRefMathSciNetGoogle Scholar
  30. Gastwirth, J. and Rubin, H. (1975). The asymptotic distribution theory of the empiric cdf for mixing stochastic processes, Ann. Stat., 3, 809–824.MATHCrossRefMathSciNetGoogle Scholar
  31. Giné, E. (1996). Empirical processes and applications: an overview, Bernoulli, 2(1), 1–28.MATHMathSciNetCrossRefGoogle Scholar
  32. Giné, E. and Zinn, J. (1984). Some limit theorems for empirical processes, with discussion, Ann. Prob., 12(4), 929–998.MATHCrossRefGoogle Scholar
  33. Hall, P. (1977). Martingale invariance principles, Ann. Prob., 5(6), 875–887.MATHCrossRefGoogle Scholar
  34. Hall, P. and Heyde, C. (1980). Martingale Limit Theory and Its Applications, Academic Press, New York.Google Scholar
  35. Heyde, C. (1981). Invariance principles in statistics, Int. Stat. Rev., 49(2), 143–152.MATHMathSciNetGoogle Scholar
  36. Jain, N., Jogdeo, K., and Stout, W. (1975). Upper and lower functions for martingales and mixing processes, Ann. Prob., 3, 119–145.MATHCrossRefMathSciNetGoogle Scholar
  37. 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
  38. Kiefer, J. (1972). Skorohod embedding of multivariate rvs and the sample df, Z. Wahr. Verw. Geb., 24, 1–35.MATHCrossRefGoogle Scholar
  39. Kolmogorov, A. (1933). Izv. Akad. Nauk SSSR, 7, 363–372 (in German).Google Scholar
  40. 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.MATHCrossRefGoogle Scholar
  41. 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.MATHCrossRefGoogle Scholar
  42. Koul, H. (1977). Behavior of robust estimators in the regression model with dependent errors, Ann. Stat., 5, 681–699.MATHCrossRefMathSciNetGoogle Scholar
  43. Major, P. (1978). On the invariance principle for sums of iid random variables, J. Multivar. Anal., 8, 487–517.MATHCrossRefMathSciNetGoogle Scholar
  44. Mandrekar, V. and Rao, B.V. (1989). On a limit theorem and invariance principle for symmetric statistics, Prob. Math. Stat., 10, 271–276.MATHMathSciNetGoogle Scholar
  45. Massart, P. (1989). Strong approximation for multivariate empirical and related processes, via KMT construction, Ann. Prob., 17(1), 266–291.MATHCrossRefMathSciNetGoogle Scholar
  46. McLeish, D. (1974). Dependent central limit theorems and invariance principles, Ann. Prob., 2, 620–628.MATHCrossRefMathSciNetGoogle Scholar
  47. McLeish, D. (1975). Invariance principles for dependent variables, Z. Wahr. Verw. Geb., 3, 165–178.CrossRefGoogle Scholar
  48. Merlevéde, F., Peligrad, M., and Utev, S. (2006). Recent advances in invariance principles for stationary sequences, Prob. Surv., 3, 1–36.CrossRefGoogle Scholar
  49. Oblój, J. (2004). The Skorohod embedding problem and its offspring, Prob. Surv., 1, 321–390.CrossRefGoogle Scholar
  50. 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
  51. 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
  52. Pollard, D. (1989). Asymptotics via empirical processes, Stat. Sci., 4, 341–366.MATHCrossRefMathSciNetGoogle Scholar
  53. Pyke, R. (1984). Asymptotic results for empirical and partial sum processes: a review, Can. J. Stat., 12, 241–264.MATHMathSciNetCrossRefGoogle Scholar
  54. Révész, P. (1976). On strong approximation of the multidimensional empirical process, Ann. Prob., 4, 729–743.MATHCrossRefGoogle Scholar
  55. Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition, Proc. Natl. Acad. Sci. USA, 42, 43–47.MATHCrossRefMathSciNetGoogle Scholar
  56. Rosenblatt, M. (1972). Uniform ergodicity and strong mixing, Z. Wahr. Verw. Geb., 24, 79–84.MATHCrossRefMathSciNetGoogle Scholar
  57. Sauer, N. (1972). On the density of families of sets, J. Comb. Theory Ser. A, 13, 145–147.MATHCrossRefMathSciNetGoogle Scholar
  58. Sen, P.K. (1978). An invariance principle for linear combinations of order statistics, Z. Wahr. Verw. Geb., 42(4), 327–340.MATHCrossRefGoogle Scholar
  59. Shorack, G. and Wellner, J. (1986). Empirical Processes with Applications to Statistics, John Wiley, New York.MATHGoogle Scholar
  60. Smirnov, N. (1944). Approximate laws of distribution of random variables from empirical data, Usp. Mat. Nauk., 10, 179–206.MATHGoogle Scholar
  61. Strassen, V. (1964). An invariance principle for the law of the iterated logarithm, Z. Wahr. Verw. Geb., 3, 211–226.MATHCrossRefMathSciNetGoogle Scholar
  62. 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
  63. van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes, Springer-Verlag, New York.MATHGoogle Scholar
  64. Vapnik, V. and Chervonenkis, A. (1971). On the uniform convergence of relative frequencies of events to their probabilities, Theory Prob. Appl., 16, 264–280.CrossRefMATHGoogle Scholar
  65. Wellner, J. (1992). Empirical processes in action: a review, Int. Stat. Rev., 60(3), 247–269.MATHGoogle Scholar
  66. Whithers, C. (1981). Conditions for linear processes to be strong mixing, Z. Wahr. Verw. Geb., 57, 477–480.CrossRefGoogle Scholar
  67. Whitt, W. (1980). Some useful functions for functional limit theorems, Math. Oper. Res., 5, 67–85.MATHMathSciNetGoogle Scholar
  68. Yu, B. (1994). Rates of convergence for empirical processes of stationary mixing sequences, Ann. Prob., 22, 94–116.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Anirban DasGupta
    • 1
  1. 1.Department of StatisticsPurdue UniversityWest Lafayette

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