# Asymptotic normality: proof strategy

Given a estimator $$\hat \theta$$ of $$\theta$$, I want to show that $$\sqrt{n}(\hat\theta -\theta-B)\to N(0,V_\theta)$$ as $$n\to\infty$$, given that the limit $$V_\theta$$ exists and $$B>0$$ possibly dependent of $$n$$.

In my case, $$\hat \theta$$ is not feasible, but there is a feasible estimator, say $$\tilde\theta$$, in the oracle case.

I plan to show this convergence from $$\sqrt{n}(\hat\theta-\theta)=\sqrt{n}(\hat\theta-\tilde\theta)+\sqrt{n}([\tilde\theta-E\tilde\theta]+[E\tilde\theta-\theta]-B)$$ by proving the following results:

1. $$\sqrt{n}(\hat\theta-\tilde\theta)=o(1)$$;
2. $$(E\tilde\theta-\theta)=B+o(1).$$;
3. $$\sqrt{n}(\tilde\theta-E\tilde\theta)\to N(0,V_\theta)$$

Do you agree with this?

Proving 2. does not prove that $$\sqrt{n}\cdot o(1) \to 0$$.
$$(E\tilde\theta-\theta)=B+o(n^{-1/2})$$
• Yes! I noticed that. Have you ever worked with Local linear estimator? In my case, the third term (in third step above) is $\sqrt{nh}\sum_{i}^n W_i(x)\epsilon_i$ where $W_n(x)$ is the local linear weight at $x$, $h$ the bandwidth and $\epsilon$ the error process. Also, $V_\theta=Var((\frac{h}{n})^{1/2}\sum_i K((i/n-x)/h))\epsilon_i$. I'm struggling to show this convergence in distribution because there is a term that is diverging in my derivations. – Celine Harumi Dec 20 '19 at 22:19