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June 2013 On the number of turns in reduced random lattice paths
Yunjiang Jiang, Weijun Xu
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J. Appl. Probab. 50(2): 499-515 (June 2013). DOI: 10.1239/jap/1371648957

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.

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Yunjiang Jiang. Weijun Xu. "On the number of turns in reduced random lattice paths." J. Appl. Probab. 50 (2) 499 - 515, June 2013. https://doi.org/10.1239/jap/1371648957

Information

Published: June 2013
First available in Project Euclid: 19 June 2013

zbMATH: 1316.60065
MathSciNet: MR3102496
Digital Object Identifier: 10.1239/jap/1371648957

Subjects:
Primary: 60F10, 60F17, 60G50

Rights: Copyright © 2013 Applied Probability Trust

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Vol.50 • No. 2 • June 2013
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