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


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

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

First available in Project Euclid: 19 June 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Signature of a path reduced word number of turns


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.

Export citation


  • Chen, K.-T. (1957). Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula. Ann. Math. 65, 163–178.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38). Springer, New York.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.
  • 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.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
  • 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.