Journal of Applied Probability
- J. Appl. Probab.
- Volume 50, Number 2 (2013), 499-515.
On the number of turns in reduced random lattice paths
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.
J. Appl. Probab., Volume 50, Number 2 (2013), 499-515.
First available in Project Euclid: 19 June 2013
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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