# Let $X_1,…, X_n$ be a sample from the normal mixture $(1-p)N(0, 1)+pN(\theta, 1)$

Let $$Z_i(\theta)=e^{X_i \theta-\frac{1}{2}\theta^2}-1$$. Is $$\sum Z_i(\theta)=O_p(\sqrt(n))$$, $$\sum Z^2_i(\theta)=O_p(1)$$ and $$\sum Z^3_i(\theta)=O_p(1)$$ and why?