As exercise 9.14 of "Measure theory and probability theory" by Krishna and Soumendra, I'm trying to demonstrate the delta method.
We have a random variable sequence $X_n$, a divergent sequence of real number $a_n \rightarrow +\infty$ and a function $H$ differentiable in $\theta$, so $H'(\theta)=c$, all such that:
\begin{gather} a_n \left( X_n - \theta\right) \longrightarrow^d Z \ \ \ \text{(converge in distribution)} \end{gather}
I want to show that
\begin{gather} a_n \left( H(X_n) - H(\theta) \right) \longrightarrow^d cZ \ \ \ \end{gather}
The book suggest to use the Taylor's expansion \begin{gather} H(X)-H(\theta) = c(X - \theta) + R(X)(X - \theta) \end{gather}
where $R(x)\rightarrow0$. Now considering the claim
\begin{gather} a_n \left( c(X_n - \theta) + R(X_n)(X_n - \theta) \right) \longrightarrow^d cZ \ \ \ \end{gather}
it's clear that $a_nc(X_n - \theta)$ tends to $cZ$ by Slutsky theorem but how to show that the remainder part tends to $0$? The book seems to suggest that $R(X_n)$ may be stochastically bounded that would be sufficient, but how to demonstrate it?
Here (in Italian, as you are) you can find the complete proof, but I want suggest another approach:
Using Cramér Rao's inequality we have, in general, the following lower bound for Variance of estimators
$$\mathbb{V}[T]\geq \frac{\Big[\frac{\partial}{\partial\theta}\mathbb{E}_{\theta}[T]\Big]^2}{nI(\theta)}$$
Where $I(\theta)$ is the Fischer Information
If we consider $T$ restricted to the class of unbiased estimator for
$$\mathbb{V}[T]\geq \frac{1}{nI(\theta)}$$
$$\mathbb{V}[T]\geq \frac{\Big[g'(\theta\Big]^2}{nI(\theta)}=\mathbb{V}[\hat{\theta}]\times \Big[g'(\theta)\Big]^2$$