I'm learning about the LLN and CLT for the first time and I'm having some trouble. I've read through other posts but I have a quirky (likely dumb) question about denominators...
The Strong LLN says for $X_1,...,X_n$ iid with common mean $\mu$,
$$P \left( \lim_{n \rightarrow \infty} \frac{X_1 + ... + X_n }{n} = \mu \right) = 1$$
My question 1: Instead of dividing by $n$, is it true that if I divide by a function of $n$, say $f(n) = Cn^k$ where $k \ge 1$ then the limit will converge to some constant always? (not necessarily $\mu$)
$$P \left( \lim_{n \rightarrow \infty} \frac{X_1 + ... + X_n }{f(n)} = \text{some constant} \right) = 1$$
I believe this is true because $Var\left(\frac{S_n}{f(n)}\right) = \frac{n \sigma^2}{(f(n))^2}$ and as long as $f(n)$ grows as fast as $n$ then the variance will go to zero as $n$ goes to infinity.
My question 2: If instead of dividing by $n$ in the Strong LLN, if I divide by $ \sigma\sqrt{n}$ then is it true that
$$P \left( \lim_{n \rightarrow \infty} \frac{X_1 + ... + X_n }{\sigma \sqrt{n}} = \text{some constant} \right) \neq 1$$
and it won't "converge" to a constant because the RV $\frac{S_n}{\sigma \sqrt{n}}$ is a Normal RV with variance $1$.
My question 3: It seems to me what you divide $S_n$ by is very important and can lead to very different results like the LLN or CLT... is that a somewhat correct way of thinking?
Thank you!!
Regarding 1 and 2: If you divide by $Cn^k$ then the limit will be $\mu/C$ (if $k=1$) or $0$ (if $k>1$). When $k<1$ the result will be $\infty$ with the same sign as $\mu$ unless $\mu=0$, in which case the situation is more complicated.
Regarding 3, yes: essentially you have different results when you measure deviation of the sample mean from the population mean in different scales relative to $n$. Under "typical" circumstances (when $X_i$ has a MGF existing in a neighborhood of $0$):