Journal of Applied Probability

On the number of turns in reduced random lattice paths

Yunjiang Jiang and Weijun Xu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the tree-reduced path of a symmetric random walk on ℤd. It is interesting to ask about the number of turns Tn in the reduced path after n steps. This question arises from inverting the signatures of lattice paths: Tn gives an upper bound of the number of terms in the signature needed to reconstruct a `random' lattice path with n steps. We show that, when n is large, the mean and variance of Tn in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for Tn, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.

Article information

Source
J. Appl. Probab., Volume 50, Number 2 (2013), 499-515.

Dates
First available in Project Euclid: 19 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1371648957

Digital Object Identifier
doi:10.1239/jap/1371648957

Mathematical Reviews number (MathSciNet)
MR3102496

Zentralblatt MATH identifier
1316.60065

Subjects
Primary: 60F10: Large deviations 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks

Keywords
Signature of a path reduced word number of turns

Citation

Jiang, Yunjiang; Xu, Weijun. On the number of turns in reduced random lattice paths. J. Appl. Probab. 50 (2013), no. 2, 499--515. doi:10.1239/jap/1371648957. https://projecteuclid.org/euclid.jap/1371648957


Export citation

References

  • Chen, K.-T. (1957). Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula. Ann. Math. 65, 163–178.
    Mathematical Reviews (MathSciNet): MR85251
    Zentralblatt MATH: 0077.25301
    Digital Object Identifier: doi:10.2307/1969671
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38). Springer, New York.
    Mathematical Reviews (MathSciNet): MR1619036
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.
    Mathematical Reviews (MathSciNet): MR228020
  • Hambly, B. and Lyons, T. (2010). Uniqueness for the signature of a path of bounded variation and the reduced path group. Ann. Math. 171, 109–167.
    Mathematical Reviews (MathSciNet): MR2630037
    Zentralblatt MATH: 05712725
    Digital Object Identifier: doi:10.4007/annals.2010.171.109
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1876169
  • Lyons, T. J. and Xu, W. (2011). Inversion of signature for paths of bounded variation. In preparation.
  • Orey, S. (1958). A central limit theorem for $m$-dependent random variables. Duke Math. J. 25, 543–546.
    Mathematical Reviews (MathSciNet): MR97841
    Zentralblatt MATH: 0107.13403
    Digital Object Identifier: doi:10.1215/S0012-7094-58-02548-1
    Project Euclid: euclid.dmj/1077468187

Applied Probability Trust

Journal of Applied Probability
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more