Approximating the log of a chi-square distribution

Question

I'm trying to solve this problem:

Let $X_1,...,X_n$ be a random sample from a $N(0,\sigma^2)$ distribution. Let $\bar{X}$ be the sample mean and let $S$ be the sample second moment $\sum X_i^2/n$. Using asymptotic theory, find an approximation to the distribution of each of the following statistics.

(a) $S$

(b) $\log S$

I can see that since $S = \frac{\sigma^2}{n}\sum Z_i^2$ for $Z_i=\frac{X_i}{\sigma} \sim N(0,1)$, $\frac{n}{\sigma^2}S\sim\chi^2(n)$ (am I right in solving (a) this way?)

But I can't figure out how to approximate the distribution of $\log S$.

I am new in learning asymptotic theory, sorry if this is a basic question or have been answered before, but I couldn't find how to solve this. Thanks in advance.

Answer 1

am I right in solving (a) this way?)

No. They are explicitly asking you to solve it with asymptotic theory.

(a) Start from the following hint:

note that $S=\frac{\Sigma_i X_i^2}{n}$ is ML Estimator for $\sigma^2$ thus to find its asymptotic law start from the properties of ML Estimators...

(b) $log S$ is a function of MLE thus to find its asymptotic law you can use Delta Method

Note that "Delta Method" is the same as applying Cramér Rao inequality for $g(\theta)$