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On the Number of Turns in Reduced Random Lattice Paths

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Yunjiang Jiang  and
Weijun Xu
Yunjiang Jiang*
Affiliation:
Stanford University
Weijun Xu*
Affiliation:
University of Oxford
*
∗ Postal address: Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, USA. Email address: [email protected]
∗∗ Postal address: Mathematical and Oxford-Man Institutes, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK. Email address: [email protected]
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Abstract

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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.

Keywords

Signature of a pathreduced wordnumber of turns

MSC classification

Primary: 60F10: Large deviations 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 499 - 515
Copyright
© Applied Probability Trust 

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