Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Invariance principles for partial sum processes and empirical processes indexed by sets
Download PDF
Download PDF
  • Published: March 1986

Invariance principles for partial sum processes and empirical processes indexed by sets

  • Gregory J. Morrow1 &
  • Walter Philipp2 

Probability Theory and Related Fields volume 73, pages 11–42 (1986)Cite this article

  • 114 Accesses

  • 5 Citations

  • Metrics details

Summary

Let |x j ,j∈ℕq} be independent identically formed random elements with values in a Banach spaceB, indexed byq-tuples of positive integers. We obtain the almost sure approximation as well as the approximation in probability of the partial sum process\(\{ \sum\limits_{j \in nA} {x_j ,A \in A\} } \) by a partial sum process\(\{ \sum\limits_{j \in nA} {y_j ,A \in A\} } \) asn→∞ uniformly over all setsA in a certain class\(A\) of subsets of theq-dimensional unit cube. Here |y j ,j∈ℕq} are independent identically distributed Gaussian random variables. These results are then applied to obtain the approximation of empirical processes over sets and indexed by sets by Gaussian partial sum processes indexed by sets.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bass, Richard F., Pyke, Ronald: Functional law of the iterated logarithm and uniform central limit theorem for partial-sum processes indexed by sets. Ann. Probab.12, 13–34 (1984)

    Google Scholar 

  2. Berkes, I., Morrow, G.J.: Strong invariance principles for mixing random fields. Z. Wahrscheinlichkeitstheor. Verw. Geb.57, 15–37 (1981)

    Google Scholar 

  3. Berkes, István, Philipp, Walter: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab.7, 29–54 (1979)

    Google Scholar 

  4. Berkes, I., Philipp, W.: An almost sure invariance principle for the empirical distribution function of mixing random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.41, 115–137 (1977)

    Google Scholar 

  5. Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

    Google Scholar 

  6. Bhattacharya, R.N.: Personal communication (1978)

  7. Carmona, R., Kono, N.: Convergence en loi et lois du logarithme itéré pour les vecteurs gaussiens. Z. Wahrscheinlichkeitstheor. Verw. Geb.36, 241–267 (1976)

    Google Scholar 

  8. Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probab.1, 66–103 (1973)

    Google Scholar 

  9. Dudley, R.M.: Empirical and Poisson processes on classes of sets or functions too large for central limit theorems. Z. Wahrscheinlichkeitstheor. Verw. Geb.61, 355–368 (1982)

    Google Scholar 

  10. Dudley, R.M.: A course in empirical processes. Lecture Notes in Mathematics1097, pp. 1–142 (1984)

    Google Scholar 

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

    Google Scholar 

  12. Erickson, R.V.: Lipschitz smoothness and convergence with applications to the central limit theorem for summation processes. Ann. Probab.9, 831–851 (1981)

    Google Scholar 

  13. Fernique, Xavier: Intégrabilité des vecteurs gaussiens. C.R. Acad. Sci. Paris, Ser. A270, A1698–1699 (1970)

    Google Scholar 

  14. Giné, E., Zinn, J.: Some limit theorems for empirical processes. Ann. Probab.12, 929–989 (1984)

    Google Scholar 

  15. Gnedenko, B.W., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Reading, Mass.: Addison-Wesley 1968

    Google Scholar 

  16. Jain, N.C., Marcus, M.B.: Continuity of subgaussian processes, Probability on Banach spaces. (Advances in Probability and Related Topics, Vol. 4, J. Kuelbs, ed.) pp. 81–196. New York-Basel: Marcel Dekker 1978

    Google Scholar 

  17. Kuelbs, J., LePage, R.: The law of the iterated logarithm for Brownian motion in a Banach space. Trans. Ann. Math. Soc.185, 253–264 (1973)

    Google Scholar 

  18. Morrow, G.J.: Approximation of rectangular sums ofB-valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.57, 265–291 (1981)

    Google Scholar 

  19. Parzen, E.: Probability density functionals and reproducing kernel Hilbert spaces. Proc. Sympos. Time Series Analysis, Brown University 1962, pp. 155–169. New York: Wiley 1963

    Google Scholar 

  20. Philipp, W.: Empirical distribution functions and uniform distribution mod 1, Diophantine Approximation and its Applications. Charles Osgood, ed., pp. 211–234. New York: Academic Press 1973

    Google Scholar 

  21. Philipp, W.: Weak andL p-invariance principles. Ann. Probab.8, 68–82 (1980); Correction, ibid.14, (1986) (to appear)

    Google Scholar 

  22. Pyke, R.: A uniform central limit theorem for partial-sum processes indexed by sets. Probab. Statist. and Anal. (J.F.C. Kingman and G.E.H. Reuter, ed.) London Math. Soc. Lecture Notes Series79, 219–240 (1983)

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

    Google Scholar 

  24. Wichura, M.: Inequalities with applications to the weak convergence of random processes with multidimensional time parameters. Ann. Math. Statist.40, 681–687 (1969)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Washington University, 63130, St. Louis, MO, USA

    Gregory J. Morrow

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

    Walter Philipp

Authors
  1. Gregory J. Morrow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Walter Philipp
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by grants from the National Science Foundation

This work was done while the first author was visiting Texas A & M University

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Morrow, G.J., Philipp, W. Invariance principles for partial sum processes and empirical processes indexed by sets. Probab. Th. Rel. Fields 73, 11–42 (1986). https://doi.org/10.1007/BF01845991

Download citation

  • Received: 10 February 1984

  • Issue Date: March 1986

  • DOI: https://doi.org/10.1007/BF01845991

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Positive Integer
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Unit Cube
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature