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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!!

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  • $\begingroup$ Please also quote the conditions on the random variables. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Oct 1 '18 at 20:49
  • $\begingroup$ Ah yes the $X_i$'s are all iid. I will edit it. $\endgroup$ – HJ_beginner Oct 1 '18 at 20:50
  • $\begingroup$ That's still incomplete. Take $\Omega = \Bbb{N}$ and $X_i = id_{\Bbb N}$ to see this. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Oct 1 '18 at 20:51
  • $\begingroup$ @GNUSupporter8964民主女神地下教會 hmmm... well I mean whatever conditions that they introduce for beginners when learning the LLNs... sorry, unfortuantely I don't know the notation in what you wrote, $X_i = id_{\Bbb N}$ - it seems $X_2 = 2d_{\Bbb N} \neq 3d_{\Bbb N} = X_3$ and so they are not iid but I think I just don't understand the notation $\endgroup$ – HJ_beginner Oct 1 '18 at 20:55
  • $\begingroup$ I mean the identity function on N. $X_i(w) = w$ for all w,i in N. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Oct 1 '18 at 20:57
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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 $0$):

  • $O(1)$ scale fluctuations asymptotically vanish (this is LLN)
  • $O(n^{-1/2})$ scale fluctuations persist with probability strictly between $0$ and $1$ (this is CLT)
  • $O(1)$ scale fluctuations decay exponentially fast in probability (this is Cramer's theorem, a type of large deviation principle).
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  • $\begingroup$ Thanks for your help. Before I learned about the LLN/CLT, I divided RVs like $S_n$ by constants all the time and it wasn't a big deal. The result would still be a RV and what the constant actually was would not dramatically change the outcome. But for LLN/CLT it seems what you divide $S_n$ by as $n$ goes to infinity has really big impact... $\endgroup$ – HJ_beginner Oct 1 '18 at 21:15
  • $\begingroup$ by the way should the third bullet point maybe be $O(n^k)$ for $k>1$? Again thanks for your help. $\endgroup$ – HJ_beginner Oct 1 '18 at 21:31
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    $\begingroup$ @HJ_beginner These are fluctuations of the sample mean around the population mean. The third bullet point is just a more precise version of the first one (but with stronger hypotheses being required): it gives a quantitative estimate of the convergence rate of $P(|\overline{X}-\mu|>\epsilon)$. $\endgroup$ – Ian Oct 1 '18 at 21:46
  • $\begingroup$ ahh thanks for the clarification!! $\endgroup$ – HJ_beginner Oct 1 '18 at 23:09

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