Lithuanian Mathematical Journal

, Volume 48, Issue 2, pp 212–227 | Cite as

Convergence in law of partial sum processes in p-variation norm

  • R. Norvaiša
  • A. Račkauskas


Let X 1, X 2, … be a sequence of independent identically distributed real-valued random variables, S n be the nth partial sum process S n (t) ≔ X 1 + ⋯ X tn, t ∈ [0, 1], W be the standard Wiener process on [0, 1], and 2 < p < ∞. It is proved that n −1/2 S n converges in law to σW as n → ∞ in p-variation norm if and only if EX 1 = 0 and σ 2 = EX 1 2 < ∞. The result is applied to test the stability of a regression model.


partial sum processes p-variation convergence in law nonlinear regression 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, 1968.Google Scholar
  2. 2.
    M. Donsker, An invariance principle for certain probability limit theorems, Mem. Am. Math. Soc., 6:1–12, 1951.MathSciNetGoogle Scholar
  3. 3.
    J.L. Doob, Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Stat., 20:393–403, 1949.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R.M. Dudley, Sample functions of the gaussian process, Ann. Probab., 1:66–103, 1973.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R.M. Dudley, Universal Donsker classes and metric entropy, Ann. Probab., 15:1306–1326, 1987.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R.M. Dudley, Fréchet differentiability, p-variation and uniform Donsker classes, Ann. Probab., 20:1968–1982, 1992.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R.M. Dudley, Uniform Central Limit Theorems, Cambridge University Press, 1999.Google Scholar
  8. 8.
    R.M. Dudley, Real Analysis and Probability, Cambridge University Press, 2002.Google Scholar
  9. 9.
    R.M. Dudley and R. Norvaiša, Differentiability of Six Operators on Nonsmooth Functions and p-Variation, in Lecture Notes in Mathematics, volume 1703. Springer, 1999.Google Scholar
  10. 10.
    P. Erdös and M. Kac, On certain limit theorems in the theory of probability, Bull. Am. Math. Soc., 52:292–302, 1946.MATHCrossRefGoogle Scholar
  11. 11.
    Y.-C. Huang and R.M. Dudley, Speed of convergence of classical empirical processes in p-variation norm, Ann. Probab., 29:1625–1636, 2001.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    K. Itô and H.P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer-Verlag, 1965.Google Scholar
  13. 13.
    S.V. Kisliakov, A remark on the space of functions of bounded p-variation, Math. Nachr., 119:137–140, 1984.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A.N. Kolmogorov and V.M. Tikhomirov, ε-entropy and ε-capacity of sets in functional spaces, Usp. Mat. Nauk, 14(2):3–86, 1959.MATHMathSciNetGoogle Scholar
  15. 15.
    J. Lamperti, On convergence of stochastic processes, Trans. Am. Math. Soc., 104:430–435, 1962.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    P. Lévy, Le mouvement brownien plan, Am. J. Math., 62:487–550, 1940.CrossRefGoogle Scholar
  17. 17.
    I.B. MacNeill, Limit processes for sequences of partial sums of regression residuals, Ann. Probab., 6:695–698, 1978.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Yu.V. Prokhorov, Convergence of random processes and limit theorems in probability, Theor. Probab. Appl., 1:157–214, 1956.CrossRefGoogle Scholar
  19. 19.
    J. Qian, The p-variation of partial sum processes and the empirical process, Ann. Probab., 26:1370–1383, 1998.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    A. Račkauskas, Hölderian properties of partial sums of regression residuals, Metrika, 63:191–205, 2006.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    A. Račkauskas and C. Suquet, Necessary and sufficient condition for the functional central limit theorem in Hölder space, J. Theor. Probab., 17(1):221–243, 2004.MATHCrossRefGoogle Scholar
  22. 22.
    G.R. Shorack, Probability for Statisticians, Springer, 2000.Google Scholar
  23. 23.
    A.V. Skorokhod, Limit theorems for stochastic processes, Theor. Probab. Appl., 1:261–290, 1956.CrossRefGoogle Scholar
  24. 24.
    A.W. Van der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes with Applications to Statistics, Springer, 1996.Google Scholar
  25. 25.
    N. Wiener, Differential space, J. Math. Phys., 2:131–174, 1923.Google Scholar
  26. 26.
    L.C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67:251–282, 1936.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania
  2. 2.Department of Econometrical AnalysisVilnius UniversityVilniusLithuania

Personalised recommendations