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
On probabilities of large deviations in Banach spaces
Download PDF
Download PDF
  • Published: June 1990

On probabilities of large deviations in Banach spaces

  • V. Bentkus1 &
  • A. Račkauskas2 

Probability Theory and Related Fields volume 86, pages 131–154 (1990)Cite this article

  • 111 Accesses

  • 15 Citations

  • Metrics details

Summary

LetX, X 1 ,X 2 ,... ∈B denote a sequence of i.i.d. random variables of a real separable Banach spaceB, Y ∈ B denote a Gaussian random variable. Suppose thatEX=EY=0 and that covariances ofX andY coincide. DenoteS n =n −1/2 (X 1 +...+X n ). We prove that under appropriate conditions

$$P\left( {||S_n || > r} \right) = P\left( {||Y|| > r} \right)\left( {1 + o\left( 1 \right)} \right) as n \to \infty $$

and give estimates of the remainder term. Applications to theω 2, the Anderson-Darling test and to the empirical characteristic functions are given.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Acosta, A. de, Araujo, A. de, Gine, E.: On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces. In: Probabilities in Banach spaces, pp. 1–68. New York Basel 1977

  • Araujo, A., Gine, E.: The central limit theorem for real and Banach valued random variables. New York Chichester Brisbane Toronto: Wiley 1980

    Google Scholar 

  • Bentkus, V.: Estimates of the proximity of sums of independent random elements in the spaceC[0,1]. Lith. Math. J.23, 1–8 (1983)

    Google Scholar 

  • Bentkus, V.: Lower bounds for the rate of convergence in the central limit theorem in Banach spaces. Lith. Math. J.25, 312–320 (1985)

    Google Scholar 

  • Bentkus, V.: On large deviations in Banach spaces (Russian). Teor. Veroyatn. Primen.31, 710–716 (1986a)

    Google Scholar 

  • Bentkus, V.: On large deviations in Banach spaces. Thesis of the Ist World Congress of the Bernoulli society, USSR Tashkent 1986, p. 757. Moscow: Nauka 1986b

    Google Scholar 

  • Bentkus, V., Račkauskas, A.: Estimates of the rate of convergence of sums of independent random variables in a Banach space. I. Lith. Math. J.22, 222–234 (1982a)

    Google Scholar 

  • Bentkus, V., Račkauskas, A.: Estimates of the rate of convergence of sums of independent random variables in a Banach space. II. Lith. Math. J.22, 344–353 (1982b)

    Google Scholar 

  • Bentkus, V., Zitikis, R.. Probabilities of large deviations for L-statistics. Lith. Math. J. (to appear 1990)

  • Bergstrom, H.: On asymptotic expansions of probability functions. Skand. Aktuarietidskr.1–2, 1–24 (1951)

    Google Scholar 

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

    Google Scholar 

  • Bolthausen, E.: Exact convergence rates in some martingale central limit theorems. Ann. Probab.10, 672–688 (1982)

    Google Scholar 

  • Butzer, P.L., Hahn, L., Westphal, U.: On the rate of approximation in the central limit theorem. J. Approximation Theory13, 327–340 (1975)

    Google Scholar 

  • Ehrhard, A.: Sur la densite du maximum d'une fonction aleatoire Gaussienne. Seminaire de probabilities XVI, Univ. Strasbourg 1980/81 (Lect. Notes Math., vol. 920, pp. 581–601) Berlin Heidelberg New York: Springer 1982

    Google Scholar 

  • Götze, F.: On the rate of convergence in the central limit theorem in Banach spaces. Ann. Probab.14, 922–942 (1986)

    Google Scholar 

  • Haeusler, E.: A note on the rate of convergence in the martingale central limit theorem. Ann. Probab.12, 635–639 (1984)

    Google Scholar 

  • Ibragimov, I.A., Linnik, J.V.: Independent and stationary connected variables (Russian). Moscow: Nauka 1965

    Google Scholar 

  • Kuelbs, J. and Kurtz, T.: Berry-Esseen estimates in Hilbert space and application to law of the iterated logarithm. Ann. Probab.2, 387–407 (1974)

    Google Scholar 

  • Osipov, L.V.: On large deviations for sums of independent random vectors (Russian). Abstracts of communications of the second Vilnius conference on probability theory and mathematical statistics, vol. 2, pp. 95–96 (1977)

    Google Scholar 

  • Paulauskas, V.I.: On the convergence rate in the central limit theorem in certain Banach spaces. Theor. Probab. Appl.21, 775–791 (1976)

    Google Scholar 

  • Paulauskas, V.I.: On the approximation of indicator functions by smooth functions in Banach spaces. Functional analysis and approximation. Basel: Birkhäuser Verlag 1981

    Google Scholar 

  • Paulauskas, V., Račkauskas, A.: Approximation theory in the central limit theorem. Exact results in Banach spaces. Norwell Laucaster Dordrecht Kluwer 1989

    Google Scholar 

  • Petrov, Yu.V.: Sums of independent random variables. Berlin Heidelberg New York: Springer 1975

    Google Scholar 

  • Rhee, W.S., Talagrand, M.: Bad rates of convergence for the central limit theorem in Hilbert space. Ann. Probab.12, 843–850 (1984)

    Google Scholar 

  • Rudzkis, R., Saulis, L., Statulevičius, V.A.: Large deviations of sums of independent random variables. Lith. Math. J.19, 118–125 (1979)

    Google Scholar 

  • Sazonov, V.V.: Normal approximation-some recent advances. (Lect. Notes Math., vol. 879, pp. 1–105) Berlin Heidelberg, New York: Springer 1981

    Google Scholar 

  • Tsirel'son, B.S.: The density of the maximum of a Gaussian process. Theor. Probab. Appl.20, 847–855 (1975)

    Google Scholar 

  • Yurinskii, V.V.: Exponential inequalities for sums of random vectors. J. Multivar. Anal.6, 473–499 (1976)

    Google Scholar 

  • Zolotarev, V.M.: Approximation of distributions of sums of independent random variables with values in infinite-dimensional spaces. Teor. Veroyatn. Primen.21, 741–757 (1976)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Cybernetics of Academy of Sciences Lithuanian, Akademijos 4, 232600, Vilnius, Lithuania

    V. Bentkus

  2. Vilnius University, Naugarduko 24, 232006, Vilnius, Lithuania

    A. Račkauskas

Authors
  1. V. Bentkus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Račkauskas
    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

Bentkus, V., Račkauskas, A. On probabilities of large deviations in Banach spaces. Probab. Th. Rel. Fields 86, 131–154 (1990). https://doi.org/10.1007/BF01474639

Download citation

  • Received: 03 March 1988

  • Revised: 11 August 1989

  • Issue Date: June 1990

  • DOI: https://doi.org/10.1007/BF01474639

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

  • Covariance
  • Banach Space
  • Stochastic Process
  • Characteristic Function
  • Probability Theory
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