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
Exact convergence rates in strong approximation laws for large increments of partial sums
Download PDF
Download PDF
  • Published: September 1987

Exact convergence rates in strong approximation laws for large increments of partial sums

  • Paul Deheuvels1 &
  • Josef Steinebach2 

Probability Theory and Related Fields volume 76, pages 369–393 (1987)Cite this article

  • 103 Accesses

  • 19 Citations

  • Metrics details

Summary

Consider partial sumsS n of an i.i.d. sequenceX 1 X 2, ..., of centered random variables having a finite moment generating function ϕ in a neighborhood of zero. The asymptotic behaviour of\(U_n = \mathop {\max }\limits_{0 \leqq k \leqq n - b_n } (S_{k - b_n } - S_k )\) is investigated, where 1≦b n ≦n denotes an integer sequence such thatb n /logn→∞ asn→∞. In particular, ifb n =o(logp n) asn→∞ for somep>1, the exact convergence rate ofU n /b n α n =1 +0 (1) is determined, where α n depends uponb n and the distribution ofX 1. In addition, a weak limit law forU n is derived. Finally, it is shown how strong invariance takes over if\(\mathop {\lim }\limits_{n \to \infty }\) b n (loglogn)2/log3 n=∞.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Chung, K.L., Erdös, P.: On the application of the Borel-Cantelli lemma. Trans. Am. Math. Soc.72, 179–186 (1952)

    Google Scholar 

  • Csörgö, S.: Erdös-Rényi laws. Ann. Statist.7, 772–787 (1979)

    Google Scholar 

  • Csörgö, M., Révész, P.: Strong approximations in probability and statistics. New York: Academic Press 1981

    Google Scholar 

  • Csörgö, M., Steinebach, J.: Improved Erdös-Rényi and strong approximation laws for increments of partial sums. Ann. Probab.9, 988–996 (1981)

    Google Scholar 

  • Deheuvels, P.: On the Erdös-Rényi theorem for random fields and sequences and its relationships with the theory of runs and spacings. Z. Wahrscheinlichkeitstheor. Verw. Geb.70, 91–115 (1985)

    Google Scholar 

  • Deheuvels, P., Devroye, L., Lynch, J.: Exact convergence rates in the limit theorems of Erdös-Rényi and Shepp. Ann. Probab.14, 209–223 (1986)

    Google Scholar 

  • Deheuvels, P., Devroye, L.: Limit laws of Erdös-Rényi-Shepp type. Ann. Probab. (to appear)

  • Deheuvels, P., Révész, P.: Weak laws for the increments of Wiener processes, Brownian bridges, empirical processes and partial sums of i.i.d.r.v's. Proc. 6th Pannonian Symp. (to appear)

  • Erdös, P., Rényi, A.: On a new law of large numbers. J. Analyse Math.23, 103–111 (1970)

    Google Scholar 

  • Hall, P., Heyde, C.C.: Martingale Limit Theory and its Application. New York: Academic Press 1980

    Google Scholar 

  • Höglund, T.: A unified formulation of the central limit theorem for small and large deviations from the mean. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 105–117 (1979)

    Google Scholar 

  • Komlós, J., Major, P., Tusnády, G. (1976) An approximation of partial sums of independent r.v's. and the sample d.f.II. Z. Wahrscheinlichkeitstheor. Verw. Geb.34, 33–58 (1976)

    Google Scholar 

  • Lukacs, E.: Characteristic functions, 2nd. edn. London: Griffin 1970

    Google Scholar 

  • Ortega, J., Wschebor, M.: On the increments of the Wiener process. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 329–339 (1984)

    Google Scholar 

  • Petrov, V.V.: On the probabilities of large deviations of sums of indpendent random variables. Theory Probab. Appl.10, 613–622 (1982)

    Google Scholar 

  • Révész, P.: On the increments of Wiener and related processes. Ann. Probab.10, 613–622 (1982)

    Google Scholar 

  • Steinebach, J.: Best convergence rates in strong approximation laws for increments of partial sums. Techn. Rep.59, Lab. Res. Stat. Probab., Carleton University, Ottawa, Canada (1985)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. L.S.T.A., Université Paris VI, T.45-55, E3, 4 Place Jussieu, F-75230, Paris Cedex 05, France

    Paul Deheuvels

  2. Fachbereich Mathematik, Philipps-Universität, Hans-Meerwein-Strasse, D-3550, Marburg, Federal Republic of Germany

    Josef Steinebach

Authors
  1. Paul Deheuvels
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josef Steinebach
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Deheuvels, P., Steinebach, J. Exact convergence rates in strong approximation laws for large increments of partial sums. Probab. Th. Rel. Fields 76, 369–393 (1987). https://doi.org/10.1007/BF01297492

Download citation

  • Received: 21 March 1986

  • Revised: 01 October 1986

  • Issue Date: September 1987

  • DOI: https://doi.org/10.1007/BF01297492

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

  • Generate Function
  • Stochastic Process
  • Asymptotic Behaviour
  • Probability Theory
  • Mathematical Biology
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