On the number of turns in reduced random lattice paths

Abstract

We consider the tree-reduced path of a symmetric random walk on ℤ^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 the signatures of lattice paths: T_n 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 T_n in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for T_n, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.