Journal of Applied Probability

On the number of turns in reduced random lattice paths

Yunjiang Jiang and Weijun Xu

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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


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