On Number of Turns in Reduced Random Lattice Paths

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.