On Number of Turns in Reduced Random Lattice Paths

Details On Number of Turns in Reduced Random Lattice Paths On Number of Turns in Reduced Random Lattice Paths

2010-07-20

arxiv.org/abs/1007.3507v3

Mathematics

Probability

15 pages

Scientific paper

We consider the tree-reduced path of symmetric random walk on $\ZZ^{d}$. It is interesting to ask about the number of turns $T_n$ in the reduced path after $n$ steps. This question arises from inverting signature for lattice paths. We show that, when $n$ is large, the mean and variance of $T_n$ have the same order as $n$, while the second order terms are O(1). We then use these estimates to obtain limit theorems for $T_n$. Similar results hold for any other finite patterns as well.

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