On Number of Turns in Reduced Random Lattice Paths

TL;DR

This paper investigates the number of turns in the reduced path of a symmetric random walk on integer lattices, deriving asymptotic mean, variance, and limit theorems, with implications for inverting signatures of lattice paths.

Contribution

It provides new asymptotic results and limit theorems for the number of turns in reduced lattice paths, extending understanding of path complexity.

Findings

Mean and variance of turns grow linearly with n

Second order terms in estimates are bounded by O(1)

Limit theorems describe the distribution of turns for large n

Abstract

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.