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Selected Works of R.M. Dudley pp 301–344Cite as

Invariance Principles for Sums of Banach Space Valued Random Elements and Empirical Processes

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Part of the book series: Selected Works in Probability and Statistics ((SWPS))

Summary

Almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable Banach space. It is then shown that empirical processes readily fit into this general framework. Thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes.

Both author were supported by NSF grants. This works was done while the second author was visiting the M.I.T. Mathematics Department

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References

  1. Acosta, Alejandro de: Existence and convergence of probability measures in Banach spaces. Trans. Amer. Math. Soc. 152, 273–298 (1970)

    MATH  MathSciNet  Google Scholar 

  2. Berkes, István, Morrow, G.: Strong invariance principles for mixing random fields. Z. Wahrscheinlichkeitstheorie verw. Geb. 57, 15–37 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berkes, I., Philipp, Walter: An almost sure invariance principle for the empirical distribution function of mixing random variables. Z. Wahrscheinlichkeitstheorie verw. Geb. 41, 115–137 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berkes, I., Philipp, Walter: Approximation theorems for independent and weakly dependent random vectors. Ann. Probability 7, 29–54 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  5. Billingsley, Patrick: Convergence of Probability Measures. New York: Wiley 1968

    MATH  Google Scholar 

  6. Borisov, I.S.: Some limit theorems for empirical distributions. Abstracts of Reports. Third Vilnius Conf. Probability. Theory Math. Statist. I, 71–72 (1981)

    Google Scholar 

  7. Bourbaki, N.: Eléments de math. Première partie, livre VI, Intégration. Paris: Hermann 1959

    Google Scholar 

  8. Breiman, Leo: Probability. Reading Mass.: Addison-Wesley 1968

    MATH  Google Scholar 

  9. Bronštein, E.M.: ε-entropy of convex sets and functions. Siberian Math. J. 17, 393–398 (transl. from Sibirsk. Mat. Ž. 17, 508–514) (1976)

    Article  Google Scholar 

  10. Dehling, Herold: Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Wahrscheinlichkeitstheorie verw. Geb., to appear (1983)

    Google Scholar 

  11. Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approximation Th. 10, 227–236 (1974); Correction, ibid. 26, 192–193 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probability 6, 899–929 (1978); Correction, ibid. 7, 909–911 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dudley, R.M.: Lower layers in ℝ2 and convex sets in ℝ3 are not GB classes. Probability in Banach Spaces II (Proc. Conf. Oberwolfach, 1978). Lecture Notes in Math. 709, 97–102. Berlin-Heidelberg-New York: Springer 1979

    Google Scholar 

  14. Dudley, R.M.: Donsker classes of functions, Statistics and Related Topics (Proc. Symp. Ottawa, 1980), pages 341–352. New York Amsterdam: North-Holland 1981

    Google Scholar 

  15. Dudley, R.M.: Vapnik-Červonenkis Donsker classes of functions. Aspects statistiques et aspects physiques des processus gaussiens (Proc. Colloque C.N.R.S. St.-Flour, 1980), C.N.R.S., Paris, 251–269 (1981b)

    Google Scholar 

  16. Durst, Mark, Dudley, R.: Empirical processes, Vapnik-Chervonenkis classes and Poisson processes. Probability and Mathematical Statistics (Wrocław) 1, 109–115 (1981)

    MathSciNet  Google Scholar 

  17. Eames, W., May, L.E.: Measurable cover functions. Canad. Math. Bull. 10, 519–523 (1967)

    MATH  MathSciNet  Google Scholar 

  18. Enflo, Per: A counterexample to the approximation problem in Banach spaces. Acta Math. 130, 309–317 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fernique, Xavier: Intégrabilité des vecteurs gaussiens. C.R. Acad. Sci. Paris Sér. A 270, A 1698–1699 (1970)

    MATH  MathSciNet  Google Scholar 

  20. Fernique, X.: Régularité de processus gaussiens. Invent. Math. 12, 304–320 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  21. Freedman, D.: Brownian motion and diffusion. San Francisco; Holden-Day 1971

    MATH  Google Scholar 

  22. Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables (transl. K.L. Chung). Reading, Mass.: Addison-Wesley 1949, 1954, 1968

    Google Scholar 

  23. Goodman, Victor, Kuelbs, James, Zinn, Joel: Some results on the LIL in Banach space with applications to weighted empirical processes. Ann. Probability 9, 713–752 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hartman, Philip, Wintner, Aurel: On the law of the iterated logarithm. Amer. J. Math. 63, 169–176 (1941)

    Article  MathSciNet  Google Scholar 

  25. Heinkel, B.: Relation entre théorème central-limite et loi du logarithme itéré dans les espaces de Banach. Z. Wahrscheinlichkeitstheorie verw. Geb. 49, 211–220 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hoffmann-Jørgensen, J.: Sums of independent Banach space valued random variables. Studia Math. 52, 159–186 (1974)

    MathSciNet  Google Scholar 

  27. Ibero, M.: Approximation forte du processus empirique fonctionnel multidimensionnel. Bull. Sci. Math. 103, 409–422 (1979a)

    MATH  MathSciNet  Google Scholar 

  28. Ibero, M.: Intégrale stochastique par rapport au processus empirique multidimensionnel. C.R. Acad. Sci. Paris Sér. A 288, A 165–168 (1979b)

    MATH  MathSciNet  Google Scholar 

  29. Ionescu Tulcea, A., and C.: On the lifting property I.J. Math. Anal. Appl. 3, 537–546 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ionescu Tulcea, A. and C.: On the lifting property II: Representation of linear operators on spaces \(L_{E^{r}}, 1 \leqq r < \infty\). J. Math. Mech. (Indiana) 11, 773–795 (1962)

    MATH  MathSciNet  Google Scholar 

  31. Jain, N.C., Marcus, M.B.: Integrability of infinite sums of independent vector-valued random variables. Trans. Amer. Math. Soc. 212, 1–36 (1975)

    MATH  MathSciNet  Google Scholar 

  32. Jain, N.C.: Central limit theorem in a Banach space, Probability in Banach Spaces (Proc. Conf. Oberwolfach, 1975). Lecture Notes in Math. 526, 113–130. Berlin-Heidelberg-New York: Springer 1976

    Google Scholar 

  33. Jain, N.C.: An example concerning CLT and LIL in Banach space. Ann. Probability 4, 690–694 (1976b)

    Article  MATH  MathSciNet  Google Scholar 

  34. James, B.R.: A functional law of the iterated logarithm for weighted empirical distributions. Ann. Probability 3, 762–772 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kahane, J.-P.: Some random series of functions. Lexington, Mass.: D.C. Heath 1968

    MATH  Google Scholar 

  36. Kiefer, J.: Skorohod embedding of multivariate rv’s, and the sample df. Z. Wahrscheinlich-keitstheorie verw. Geb. 24, 1–35 (1972)

    Article  MATH  Google Scholar 

  37. Kolčinski, V.I.: On the central limit theorem for empirical measures. Teor. Verojatnost. i. Mat. Statist. (Kiev) 1981, no. 24, 63–75 (in Russian). Theory Probability and Math. Statist. no. 24, 71–82 (1981a)

    Google Scholar 

  38. Kolčinski, V.I.: On the law of the iterated logarithm in Strassen’s form for empirical measures. Teor. Verojatnost. i. Mat. Statist. (Kiev) 1981, no. 25, 40–47. Theory Probability and Math. Statist. no. 25, 43–50 (1981b)

    Google Scholar 

  39. Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV’ – s, and the sample DF. I. Z. Wahrscheinlichkeitstheorie verw. Geb. 32, 111–131 (1975)

    Article  MATH  Google Scholar 

  40. Kuelbs, J.: A strong convergence theorem for Banach space valued random variables. Ann. Probability 4, 744–771 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  41. Kuelbs, J.: Kolmogorov’s law of the iterated logarithm for Banach space valued random variables. Illinois J. Math. 21, 784–800 (1977)

    MATH  MathSciNet  Google Scholar 

  42. Kuelbs, J., Zinn, J.: Some stability results for vector valued random variables. Ann. Probability 7, 75–84 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  43. Kuelbs, J., Dudley, R.M.: Log log laws for empirical measures. Ann. Probability 8, 405–418 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  44. Kuelbs, J., Philipp, Walter: Almost sure invariance principles for partial sums of mixing B-valued random variables. Ann. Probability 8, 1003–1036 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  45. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I. Berlin-Heidelberg-New York: Springer 1977

    MATH  Google Scholar 

  46. Major, Péter: Approximation of partial sums of i.i.d. rv’s when the summands have only two moments. Z. Wahrscheinlichkeitstheorie verw. Geb. 35, 221–229 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  47. Marcus, Michael, B., Philipp, Walter: Almost sure invariance principles for sums of B-valued random variables with applications to random Fourier series and the empirical characteristic process. Trans. Amer. Math. Soc. 269, 67–90 (1982)

    MATH  MathSciNet  Google Scholar 

  48. Marcus, M.B., Shepp, L.A.: Sample behavior of Gaussian processes. Proc. 6th Berkeley Sympos. Math. Statist. Probab. 2, 423–442. Univ. Calif. Press (1971)

    Google Scholar 

  49. May, L.E.: Separation of functions. Canad. Math. Bull. 16, 245–250 (1973)

    MATH  MathSciNet  Google Scholar 

  50. O’Reilly, Neville E.: On the convergence of empirical processes in sup-norm metrics. Ann. Probability 2, 642–651 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  51. Parthasarathy, K.R.: Probability measures on metric spaces. New York: Academic Press 1967

    MATH  Google Scholar 

  52. Philipp, Walter: Almost sure invariance principles for sums of B-valued random variables, Probability in Banach Spaces II (Proc. Conf. Oberwolfach, 1978). Lecture Notes in Math. 709, 171–193. Berlin-Heidelberg-New York: Springer 1979

    Google Scholar 

  53. Philipp, Walter: Weak and L p-invariance principles for sums of B-valued random variables. Ann. Probability 8, 68–82 (1980); Correction, ibid. (to appear)

    Article  MATH  MathSciNet  Google Scholar 

  54. Philipp, Walter, Pinzur, Laurence: Almost sure approximation theorems for the multivariate empirical process. Z. Wahrscheinlichkeitstheorie verw. Geb. 54, 1–13 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  55. Pisier, G.: Le théorème de la limite centrale et la loi du logarithme itéré dans les espaces de Banach. Séminaire Maurey-Schwartz 1975–76, Exposé IV, École Polytechnique, Paris (1975)

    Google Scholar 

  56. Pollard, D.: Limit theorems for empirical processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 57, 181–195 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  57. Pollard, D.: A central limit theorem for empirical processes. [To appear in J. Australian Math. Soc. Ser. A]

    Google Scholar 

  58. Prohorov, Yu. V.: Convergence of random processes and limit theorems in probability theory. Theor. Probability Appl. 1, 157–214 (1956)

    Article  MathSciNet  Google Scholar 

  59. Révész, Pál: On strong approximation of the multidimensional empirical process. Ann. Probability 4, 729–743 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  60. Senatov, V.V.: Some estimates of the rate of convergence in the multidimensional central limit theorem. Soviet Math. Doklady 22, 462–466 = Dokl. Akad. Nauk SSSR 254 no. 4, 809–812 (1980)

    MathSciNet  Google Scholar 

  61. Shortt, R.M.: Existence of laws with given marginals and specified support. Ph.D. thesis, M.I.T. (1982)

    Google Scholar 

  62. Skorohod, A.V.: Limit theorems for stochastic processes. Theor. Probability Appl. 1, 261–290 (1956)

    Article  MathSciNet  Google Scholar 

  63. Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  64. Vorob’ev, N.N.: Consistent families of measures and their extensions. Theor. Probability Appl. 7, 147–163 (1962)

    Article  Google Scholar 

  65. Vulikh, B.Z.: Introduction to the theory of partially ordered spaces (transl. by L.F. Boron). Gröningen: Walters-Noordhoff 1967

    Google Scholar 

  66. Wenocur, R.S., Dudley, R.M.: Some special Vapnik-Chervonenkis classes. Discrete Math. 33, 313–318 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  67. Yurinskii, V.V.: On the error of the Gaussian approximation for convolutions. Theor. Probability Appl. 22, 236–247 (1977)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. M.I.T., Room 2-245, Cambridge, MA, 02139, USA

    R. M. Dudley

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Walter Philipp

Authors
  1. R. M. Dudley
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  2. Walter Philipp
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Editors and Affiliations

  1. , Mathematics, University of Connecticut, Auditorium Road 196, Storrs, 06269-3009, USA

    Evarist Giné

  2. , Mathematics, Georgia Institute of Technology, Atlanta, 30332-0160, USA

    Vladimir Koltchinskii

  3. Inst. Mathematics & Informatics, Akademijos 4, Vilnius, LT-08663, Lithuania

    Rimas Norvaisa

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Dudley, R.M., Philipp, W. (2010). Invariance Principles for Sums of Banach Space Valued Random Elements and Empirical Processes. In: Giné, E., Koltchinskii, V., Norvaisa, R. (eds) Selected Works of R.M. Dudley. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5821-1_18

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