Skip to main content
SpringerLink
Log in

KMT coupling for random walk bridges

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

In this paper we prove an analogue of the Komlós–Major–Tusnády (KMT) embedding theorem for random walk bridges. The random bridges we consider are constructed through random walks with i.i.d jumps that are conditioned on the locations of their endpoints. We prove that such bridges can be strongly coupled to Brownian bridges of appropriate variance when the jumps are either continuous or integer valued under some mild technical assumptions on the jump distributions. Our arguments follow a similar dyadic scheme to KMT’s original proof, but they require more refined estimates and stronger assumptions necessitated by the endpoint conditioning. In particular, our result does not follow from the KMT embedding theorem, which we illustrate via a counterexample.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1970)

    MATH  Google Scholar 

  2. Bártfai, P.: Die Bestimmung der zu einem wiederkehrenden Prozeß gehörenden Verteilungsfunktion aus den mit fehlern behafteten Daten einer Einzigen Realisation. Studia Sci. Math. Hung. 1, 161–168 (1966)

    MATH  Google Scholar 

  3. Batir, N.: Inequalities for the gamma function. N. Arch. Math. 91, 554–563 (2008)

    Article  MathSciNet  Google Scholar 

  4. Berkes, I., Liu, W., Wu, W.: Komlós, Major and Tusnády approximation under dependence. Ann. Probab. 42, 794–817 (2014)

    Article  MathSciNet  Google Scholar 

  5. Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields 158, 225–400 (2014)

    Article  MathSciNet  Google Scholar 

  6. Calvert, J., Hammond, A., Hedge, M.: Brownian structure in the KPZ fixed point (2019). Preprint arXiv:1912.00992

  7. Chatterjee, S.: A new approach to strong embeddings. Probab. Theory Relat. Fields 152, 231–264 (2012)

    Article  MathSciNet  Google Scholar 

  8. Corwin, I., Dimitrov, E.: Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles. Commun. Math. Phys. 363, 435–501 (2018)

    Article  MathSciNet  Google Scholar 

  9. Corwin, I., Hammond, A.: KPZ line ensemble. Probab. Theory Relat. Fields 166, 67–185 (2016)

    Article  MathSciNet  Google Scholar 

  10. Corwin, I., O’Connell, N., Seppäläinen, T., Zygouras, N.: Tropical combinatorics and Whittaker functions. Duke Math. J. 163, 513–563 (2014)

    Article  MathSciNet  Google Scholar 

  11. Csörgő, M., Hall, P.: The Komlós, Major and Tusnády approximations and their applications. Aust. J. Stat. 26, 189–218 (1984)

    Article  Google Scholar 

  12. Csörgő, M., Révész, P.: Strong Approximations in Probability and Statitstics. Academic, New York (1981)

    MATH  Google Scholar 

  13. Dauvergne, D., Nica, M., Virág, B.: Uniform convergence to the Airy line ensemble (2019). arXiv:1907.10160

  14. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)

    Book  Google Scholar 

  15. Durrett, R.: Probability: Theory and Examples, 4th edn. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  16. Einmahl, U.: Extensions of results of Komlós, Major and Tusnády to the multivariate case. J. Multivar. Anal. 28, 20–68 (1989)

    Article  Google Scholar 

  17. Götze, F., Zaitsev, A.Y.: Bounds for the rate of strong approximation in the multidimensional invariance principle. Theory Probab. Appl. 53, 59–80 (2009)

    Article  MathSciNet  Google Scholar 

  18. Guo, B.-N., Qi, F.: Two new proofs of the complete monotonicity of a function involving the psi function. Bull. Korean Math. Soc. 47, 103–111 (2010). arXiv:0902.2520

  19. Hammond, A.: Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation (2016). arXiv:1609.029171

  20. Hammond, A.: Exponents governing the rarity of disjoint polymers in Brownian last passage percolations (2017). Preprint arXiv:1709.04110

  21. Hammond, A.: Modulus of continuity of polymer weight profiles in Brownian last passage percolation (2017). Preprint arXiv:1709.04115

  22. Hammond, A.: A patchwork quilt sewn from Brownian fabric: regularity of polymer weight profiles in Brownian last passage percolation. Forum Math. Pi 7 (2019). Preprint arXiv:1709.04113

  23. Ibragimov, I.A.: On the composition of unimodal distributions. Theory Probab. Appl. 1, 255–260 (1956)

    Article  MathSciNet  Google Scholar 

  24. Johansson, K.: Shape fluctuations and random matrices. Comm. Math. Phys. 209, 437–476 (2000)

    Article  MathSciNet  Google Scholar 

  25. Keilson, J., Gerber, H.: Some results for discrete unimodality. J. Am. Stat. Assoc. 66, 386–389 (1971)

    Article  Google Scholar 

  26. Kiefer, J.: On the deviations in the Skorokhod-Strassen approximation scheme. Z. Wahrsch. Verw. Gebiete 13, 321–332 (1969)

    Article  MathSciNet  Google Scholar 

  27. Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV’s, and the sample DF I. Z. Wahrsch. Verw. Gebiete 32, 111–131 (1975)

    Article  MathSciNet  Google Scholar 

  28. Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV’s, and the sample DF II. Z. Wahrsch. Verw. Gebiete 34, 33–58 (1976)

    Article  MathSciNet  Google Scholar 

  29. Lawler, G.F., Trujillo-Ferreras, J.A.: Random walk loop soup. Trans. Am. Math. Soc. 359, 767–787 (2007)

    Article  MathSciNet  Google Scholar 

  30. Li, X., Chen, C.-P.: Inequalities for the gamma function. J. Inequal. Pure Appl. Math. 8, 3 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Mason, D., Zhou, H.: Quantile coupling inequalities and their applications. Probab. Surv. 9, 439–479 (2012)

    Article  MathSciNet  Google Scholar 

  32. Mattner, L.: Complex differentiation under the integral. Nieuw Arch. Wiskd. 4, 32–35 (2001)

    MathSciNet  MATH  Google Scholar 

  33. Obłój, J.: The Skorokhod embedding problem and its offspring. Probab. Surv. 1, 321–392 (2004)

    Article  MathSciNet  Google Scholar 

  34. Shao, Q.M.: Strong approximation theorems for independent random variables and their applications. J. Multivar. Anal. 52, 107–130 (1995)

    Article  MathSciNet  Google Scholar 

  35. Sakhanenko, A.I.: Rate of convergence in the invariance principle for variables with exponential moments that are not identically distributed. In: Trudy Inst. Mat. SO AN SSSR 3, Nauka, Novobirsk, pp. 4–49 (1984) (in Russian)

  36. Sakhanenko, A.I.: A general estimate in the invariance principle. Sib. Math. J. 52, 696–710 (2011)

    Article  MathSciNet  Google Scholar 

  37. Shorack, G.R., Wellner, J.A.: Empirical Processes with Applications to Statistics. Wiley, New York (1986)

    MATH  Google Scholar 

  38. Skorohod, A.V.: Issledovaniya po teorii sluchainykh protsessov, (Stokhasticheskie differentsialnye uravneniya i predelnye teoremy dlya protsessov Markova) Izdat. Kiev. Univ, Kiev (1961)

  39. Skorohod, A..V.: Studies in the theory of random processes. Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading (1965)

  40. Stein, E., Shakarchi, R.: Complex Analysis. Princeton University Press, Princeton (2003)

    MATH  Google Scholar 

  41. Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. In: Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA 1965/1966), vol II, Part 1, pp. 315–343. University of California Press, Berkeley (1967)

  42. Wu, X.: Discrete Gibbsian line ensembles and weak noise scaling for directed polymers. PhD Thesis (2020). https://academiccommons.columbia.edu/doi/10.7916/d8-6re1-k703

  43. Zaitsev, A.Y.: Estimates for the quantiles of smooth conditional distributions and the multidimensional invariance principle. Sib. Math. J. 37, 706–729 (1996)

    Article  MathSciNet  Google Scholar 

  44. Zaitsev, A.Y.: Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments. ESIAM Probab. Stat. 2, 41–108 (1998)

    Article  Google Scholar 

  45. Zaitsev, A.Y.: Estimates for the strong approximation in multidimensional central limit theorem. Proc. Int. Cong. Math. III, 107–116 (2002)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are deeply grateful to Ivan Corwin for many useful suggestions and comments as well as Julien Dubedat, Alisa Knizel and Konstantin Matetski for numerous fruitful discussions. The authors are also grateful to the anonymous referee for their comments and suggestions. The first author is partially supported by the Minerva Foundation Fellowship. For the second author partial financial support was available through the NSF grants DMS:1811143, DMS:1664650 and the Minerva Foundation Summer Fellowship program.

Author information

Authors and Affiliations

  1. Columbia University, 2990 Broadway, Rm. 517, New York, NY, 10029, USA

    Evgeni Dimitrov

  2. University of Chicago, 5734 S University Ave, Eckhart Hall, Rm. 319, Chicago, IL, 60637, USA

    Xuan Wu

Authors
  1. Evgeni Dimitrov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evgeni Dimitrov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dimitrov, E., Wu, X. KMT coupling for random walk bridges. Probab. Theory Relat. Fields 179, 649–732 (2021). https://doi.org/10.1007/s00440-021-01030-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-021-01030-y

Mathematics Subject Classification

Search

Navigation