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
Asymptotic behaviour of a class of stochastic approximation procedures
Download PDF
Download PDF
  • Published: December 1986

Asymptotic behaviour of a class of stochastic approximation procedures

  • Erich Berger1 

Probability Theory and Related Fields volume 71, pages 517–552 (1986)Cite this article

  • 169 Accesses

  • 30 Citations

  • Metrics details

Summary

LetB be a real Banach space. Considering stochastic approximation procedures of Robbins-Monro and Kiefer-Wolfowitz type, one is led to recursion formulas of the form

$$U_{k + 1} = U_k - k^{ - 1} \Gamma _k U_k - k^{ - \rho } V_k - k^{ - \eta } T_k$$

(ϱ and η) positive real numbers,U k, Vk, Tk B-valued random variables, Γ k random linear operatorsB→B, where — in addition to some further assumptions —Γ k →Γ almost surely andT k→T, say, almost surely. Letting (Y k,n )k∈{1,...,n},n∈N be an array ofB-valued random variables such that the partial sums\(\sum\limits_{k = 1}^m {V_k }\) and\(\sum\limits_{k = 1}^m {Y_{k,n} }\) do not differ “too much” (uniformly inm∈{1,...,n}), we shall investigate the distance of (U m ) m∈{1,...,n} and (U m (n) ) m∈{1,...,n}, where

$$U_{k + 1}^{\left( n \right)} = U_k^{\left( n \right)} - k^{ - 1} \Gamma U_k^{\left( n \right)} - k^{ - \rho } Y_{k,n} - k^{ - \eta } T$$

fork∈{1,...,n−1}. The proofs are based on a deterministic argument that enables us to present a unified approach for proving central limit theorems, weak and almost sure invariance principles and variants of the bounded and the functional law of the iterated logarithm for the sequence (U n).

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Abdelhamid, S.N.: Transformation of observations in stochastic approximation. Ann. Statist.1, 1158–1174 (1973)

    Google Scholar 

  2. Berger, E.: A note on the invariance principle for stochastic approximation procedures in a Hilbert space. Unpublished manuscript (1979)

  3. Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968

    Google Scholar 

  4. Birnbaum, Z.W., Marshall, A.W.: Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Ann. Math. Statist.32, 687–704 (1961)

    Google Scholar 

  5. Blum, J.R.: Approximation methods which converge with probability one. Ann. Math. Statist.25, 382–386 (1954)

    Google Scholar 

  6. Borodin, A.N.: A stochastic approximation procedure in the case of weakly dependent observations. Theory Probab. Appl.24 34–52 (1979)

    Google Scholar 

  7. Chung, K.L.: On a stochastic approximation method. Ann. Math. Statist.25, 463–483 (1954)

    Google Scholar 

  8. Daleckiî, Ju.L., Kreîn, M.G.: Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs43. Providence, Rhode Island: American Math. Society 1974

    Google Scholar 

  9. Dehling, H.: Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.63, 393–432 (1983)

    Google Scholar 

  10. Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Statist.39, 1563–1572 (1968)

    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. Fabian, V.: On asymptotic normality in stochastic approximation. Ann. Math. Statist.39, 1327–1332 (1968)

    Google Scholar 

  13. Fabian, V.: Asymptotically efficient stochastic approximation; the RM case. Ann. Statist.1, 485–495 (1973)

    Google Scholar 

  14. Fabian, V.: On asymptotically efficient recursive estimation. Ann. Statist.6, 854–866 (1978)

    Google Scholar 

  15. Fabian, V.: Local asymptotic minimax properties of some recursive estimates.RM — 409. Department of Statistics and Probability. Michigan State University (1980)

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

    Google Scholar 

  17. Gänssler, P., Stute, W.: Wahrscheinlichkeitstheorie. Berlin-Heidelberg-New York: Springer 1977

    Google Scholar 

  18. Gaposhkin, V.F., Krasulina, T.P.: On the law of the iterated logarithm in stochastic approximation processes. Theory Probab. Appl.19, 844–850 (1974)

    Google Scholar 

  19. Hall, P.G., Heyde, C.C.: On a unified approach to the law of the iterated logarithm for martingales. Bull. Aust. Math. Soc.14, 435–447 (1976)

    Google Scholar 

  20. Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Applications. New York-San Francisco-London: Academic Press 1980

    Google Scholar 

  21. Heyde, C.C.: On martingale limit theory and strong convergence results for stochastic approximation procedures. Stochastic Processes Appl.2, 371–389 (1974)

    Google Scholar 

  22. Jakubowski, A.: On limit theorems for sums of dependent Hilbert space valued random variables. In: Mathematical Statistics and Probability Theory. Proc. 6th Int. Conf., Wisla/Pol. 1978. Lect. Notes Stat.2, 178–187 (1980)

  23. Kersting, G.: Almost sure approximation of the Robbins-Monro process by sums of independent random variables. Ann. Probab.5, 954–965 (1977)

    Google Scholar 

  24. Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheor. Verw. Geb.32, 111–131 (1975)

    Google Scholar 

  25. Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV's and the sample DF. II. Z. Wahrscheinlichkeitstheor. Verw. Geb.34, 33–58 (1976)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  28. Lai, T.L., Robbins, H.: Limit theorems for weighted sums and stochastic approximation processes. Proc. Nat. Acad. Sci. U.S.A.75, 1068–1070 (1978)

    Google Scholar 

  29. Lai, T.L., Robbins, H.: Adaptive design and stochastic approximation. Ann. Statist.7, 1196–1221 (1979)

    Google Scholar 

  30. Major, P., Révész, P.: A limit theorem for the Robbins-Monro approximation. Z. Wahrscheinlichkeitstheor. Verw. Geb.27, 79–86 (1973)

    Google Scholar 

  31. Major, P.: The approximation of partial sums of independent RV's Z. Wahrscheinlichkeitstheor. Verw. Geb.35, 213–220 (1976)

    Google Scholar 

  32. Mark, G.: Log-log-Invarianzprinzipien für Prozesse der stochastischen Approximation. Mitteilungen Math. Sem. Giessen, No. 153, 87 pp. (1982)

  33. McLeish, D.L.: Invariance principles for dependent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.32, 165–178 (1975)

    Google Scholar 

  34. McLeish, D.L.: Functional and random central limit theorems for the Robbins-Monro process. J. Appl. Probab.13, 148–154 (1976)

    Google Scholar 

  35. McLeish, D.L.: On the invariance principle for nonstationary mixingales. Ann. Probab.5, 616–621 (1977)

    Google Scholar 

  36. Métivier, M.: Semimartingales: a Course on Stochastic Processes. Berlin-New York: de Gruyter 1982

    Google Scholar 

  37. Morrow, G., Philipp, W.: An almost sure invariance principle for Hilbert space valued martingales. Trans. Am. Math. Soc.273, 231–251 (1982)

    Google Scholar 

  38. Nevel'son, M.B., Has'minskiî, R.Z.: Stochastic Approximation and Recursive Estimation. Translations of Mathematical Monographs47. Providence, Rhode Island: American Math. Society 1976

    Google Scholar 

  39. Nixdorf, R.: Stochastische Approximation in Hilberträumen durch endlichdimensionale Verfahren. Mitteilungen Math. Sem. Giessen, No. 154, 78 pp. (1982)

  40. Nixdorf, R.: An invariance principle for a finite dimensional stochastic approximation method in a Hilbert space. J. Multivariate Analysis15, 252–260 (1984)

    Google Scholar 

  41. O'Reilly, N.E.: On weak convergence of empirical processes in sup-norm metrics. Ann. Probab.2, 642–651 (1974)

    Google Scholar 

  42. Pantel, M.: Adaptive Verfahren der stochastischen Approximation. Dissertation. Universität Essen — Gesamthochschule (1979)

  43. Philipp, W., Stout, W.: Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. No. 161. Providence, Rhode Island: American Math. Society 1975

    Google Scholar 

  44. Philipp, W.: Almost sure invariance principles for sums ofB-valued random variables. In: Probability in Banach Spaces II. Proc. Conf. Oberwolfach 1978. Lect. Notes in Math.709, pp. 171–193, Berlin-Heidelberg-New York: Springer 1979

    Google Scholar 

  45. Pyke, R.: Applications of almost surely convergent constructions of weakly convergent processes. In: Probability and Information Theory. Proc. Int. Symp. McMaster Univ. 1968. Lect. Notes in Math.89, 187–200. Berlin-Heidelberg-New York: Springer 1969

    Google Scholar 

  46. Ruppert, D.: Almost sure approximations to the Robbins-Monro and Kiefer-Wolfowitz processes with dependent noise. Ann. Probab.10, 178–187 (1982)

    Google Scholar 

  47. Sacks, J.: Asymptotic distribution of stochastic approximation procedures. Ann. Math. Statist.29, 373–405 (1958)

    Google Scholar 

  48. Skorohod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl.1, 261–290 (1956)

    Google Scholar 

  49. Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 211–226 (1964)

    Google Scholar 

  50. Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Sympos. Math. Statist. Probab.2, 315–344 (1965)

    Google Scholar 

  51. Venter, J.H.: An extension of the Robbins-Monro procedure. Ann. Math. Statist.38, 181–190 (1967)

    Google Scholar 

  52. Walk, H.: An invariance principle for the Robbins-Monro process in a Hilbert space. Z. Wahrscheinlichkeitstheor. Verw. Geb.39, 135–150 (1977)

    Google Scholar 

  53. Walk, H.: Sequential estimation of the solution of an integral equation in filtering theory. In: Stochastic Control Theory and Stochastic Differential Systems (eds.: M. Kohlmann, W. Vogel). Lect. Notes in Control and Information Sciences16, 598–605 (1979)

  54. Walk, H.: Martingales and the Robbins-Monro procedure inD[0, 1].J. Multivariate Anal. 8, 430–452 (1978)

    Google Scholar 

  55. Walk, H.: A functional central limit theorem for martingales inC(K) and its application to sequential estimates. J. Reine Angew. Math.314, 117–135 (1980)

    Google Scholar 

  56. Watanabe, M.: A stochastic approximation from dependent observations. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 279–292 (1983)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, D-3400, Göttingen, Germany

    Erich Berger

Authors
  1. Erich Berger
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Part of this work was done while the author was supported by Deutsche Forschungsgemeinschaft

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Berger, E. Asymptotic behaviour of a class of stochastic approximation procedures. Probab. Th. Rel. Fields 71, 517–552 (1986). https://doi.org/10.1007/BF00699040

Download citation

  • Received: 17 December 1984

  • Issue Date: December 1986

  • DOI: https://doi.org/10.1007/BF00699040

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

  • Real Number
  • Stochastic Process
  • Asymptotic Behaviour
  • Probability Theory
  • Limit Theorem
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