Is there a theorem that says that $\sqrt{n}\frac{\bar{X} - \mu}{S}$ converges in distribution to a normal as $n$ goes to infinity?

Question

Let $X$ be any distribution with defined mean, $\mu$, and standard deviation, $\sigma$. The central limit theorem says that $$ \sqrt{n}\frac{\bar{X} - \mu}{\sigma} $$ converges in distribution to a standard normal distribution. If we replace $\sigma$ by the sample standard deviation $S$, is there a theorem stating that $$ \sqrt{n}\frac{\bar{X} - \mu}{S} $$ converges in distribution to a t-distribution? Since for large $n$ a t-distribution approaches a normal, the theorem, if it exists, may state that the limit is a standard normal distribution. Hence, it would seem to me that t-distributions are not very useful - that they are useful only when $X$ is roughly a normal. Is this the case?

If it is possible, would you indicate references that containing a proof of this CLT when $\sigma$ is replace by $S$? Such a reference could preferably use measure theory concepts. But anything would be great to me at this point.

Answer 1

To elaborate on @cardinal 's comment, consider an i.i.d. sample of size $n$ from a random variable $X$ with some distribution, and finite moments, mean $\mu$ and standard deviation $\sigma$. Define the random variable

$$Z_n = \sqrt {n}\left(\bar X_n -\mu\right)$$ The basic Central Limit Theorem says that $$Z_n \rightarrow_{d} Z \sim N(0,\sigma^2)$$

Consider now the random variable $Y_n = \frac 1{S_n}$ where $S_n$ is the sample standard deviation of $X$.

The sample is i.i.d and so sample moments estimate consistently population moments. So

$$Y_n \rightarrow_{p} \frac 1{\sigma} $$

Enter @cardinal: Slutsky's theorem (or lemma) says, among other things, that $$ \{Z_n \rightarrow_{d} Z, Y_n\rightarrow_{p} c\} \Rightarrow Z_nY_n\rightarrow_{d} cZ$$ where $c$ is a constant. This is our case so

$$Z_nY_n = \sqrt{n}\frac{\bar{X_n} - \mu}{S_n}\rightarrow_{d} \frac 1{\sigma}Z \sim N(0,1)$$

As for the usefulness of Student's distribution, I only mention that, in its "traditional uses" related to statistical tests it still is indispensable when sample sizes are really small (and we are still confronted with such cases), but also, that it has been widely applied to model autoregressive series with (conditional) heteroskedasticity, especially in the context of Finance Econometrics, where such data arise frequently.