Skip to main content

Invariance Principle for the Random Walk Conditioned to Have Few Zeros

  • 972 Accesses

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2123)

Abstract

We consider a nearest neighbor random walk on \(\mathbb{Z}\) starting at zero, conditioned to return at zero at time 2n and to have a number z n of zeros on (0, 2n]. As \(n \rightarrow +\infty \), if \(z_{n} = o(\sqrt{n})\), we show that the rescaled random walk converges toward the Brownian excursion normalized to have unit duration. This generalizes the classical result for the case z n  ≡ 1.

Keywords

  • Conditioned random walk
  • Excursions
  • Invariance principle

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968)

    MATH  Google Scholar 

  2. F. Caravenna, L. Chaumont, An invariance principle for random walk bridges conditioned to stay positive. Electron. J. Probab. 18(60), 1–32 (2013)

    MathSciNet  Google Scholar 

  3. E. Csaki, Y. Hu, Lengths and heights of random walk excursions. Discrete Math. Theor. Comput. Sci. AC, 45–52 (2003)

    Google Scholar 

  4. G. Giacomin, Random Polymer Models (Imperial College Press, World Scientific, 2007)

    CrossRef  MATH  Google Scholar 

  5. W.D. Kaigh, An invariance principle for random walk conditioned by a late return to zero. Ann. Probab. 4, 115–121 (1976)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J.F. Le Gall, Random trees and applications. Probab. Surv. 2, 245–311 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. T. Liggett, An invariance principle for conditioned sums of independent random variables. J. Math. Mech. 18, 559–570 (1968)

    MathSciNet  MATH  Google Scholar 

  8. P. Révész, Local times and invariance. Analytic Methods in Probab. Theory. LNM 861 (Springer, New York, 1981), pp. 128–145

    Google Scholar 

  9. P. Révész, Random Walk in Random and Non-random Environments, 2nd edn. (World Scientific, Singapore, 2005)

    CrossRef  MATH  Google Scholar 

  10. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, New York, 1999)

    CrossRef  MATH  Google Scholar 

  11. R.P. Stanley, Enumerative Combinatorics, vol. 2 (Cambridge University Press, Cambridge, 1999)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Laurent Serlet .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Serlet, L. (2014). Invariance Principle for the Random Walk Conditioned to Have Few Zeros. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_19

Download citation