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In an exercise to prove the Delta Method, the author suggested using Taylor expansion, stating the following:

By Taylor’s expansion, for any $x \in \mathbb R$

$$H(x) = H(\theta) + c(x- \theta) + R(x)(x-\theta)$$

Where $R(x)\rightarrow 0$ as $x\rightarrow \theta$.

Now, I have proved that $X_n \rightarrow_p \theta$. How does one then proves that $R(X_n)\rightarrow_p 0$?

StubbornAtom
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Davi Barreira
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  • What is $c$? $H'(\theta)$? –  Aug 05 '20 at 19:45
  • Yes. I didn’t write here the complete question, sorry if that made things less clear. The only really relevant part for my question, though, is how to show that the remainder goes to zero. One does not need to worry about $H$. Cheers – Davi Barreira Aug 05 '20 at 19:48

2 Answers2

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Well... just plug into the definition.

Let $\varepsilon>0$ and pick $\delta$ such that $|x-\theta|<\delta$ implies $|R(x)|<\varepsilon$.

Then, $\{|R(X_n)|\geq \varepsilon\}\subseteq \{|X_n-\theta|\geq \delta\}$. Now, use the fact that $X_n$ converges to $\theta$ in probability.

WoolierThanThou
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For every subsequence $\{X_{n_k}\}$ there exists further sub-subsequence $\{X_{n_{k(l)}}\}$ s.t. $X_{n_{k(l)}}\to \theta$ a.s. and $R(X_{n_{k(l)}})\to 0$ a.s. Then use this result.