Is my professor wrong again?

svarog · #1 $Old$ October 26th, 2009, 10:40 AM

My professor has the annoying habit of consistently giving us problems which are stated wrong (after few hours of

you eventually figure out it's not you that's making the mistake), but this time I think it happened on the midterm. Could someone please look at this:

For what s=? does $Var(X_n) ->_p0$ ?
$X_n$ is the sample mean of n independent $X_i$ (not identically distributed), where $X_i=i^s$ with $p=1/2$ and $X_i=-i^s$ with $p=1/2$ .

Here's my reasoning. $Var(X_i)=1/2*(i^{2s}+i^{2s})=i^{2s}$ so then $Var(X_n)=Var(1/n*sum(X_i))=1/n^2*sum(Var(X_i))=$ (for i from 1 to n)
$=1/n^2*sum(i^{2s})=O_p(n^{2s}/n^2)=O_p(n^{2(s-1)})$ as n->oo
This means the highest order of n in the sequence is 2(s-1). The sequence will go to 0 as n goes to infinity (for all terms) if the highest order also goes to 0. So 2(s-1)<0, thus s<1.
However, the problem asked me to prove that s<1/2. I asked the professor to make sure my assumptions are correct; he confirmed, however the result still disagrees. To be diplomatic, I wrote the solution until the last line (with $O_p$ ), however I wrote that the conclusion is s<1/2. (just to make sure i'm wrong no matter what happens

)) Opinions?

Laurent · #2 $Old$ October 26th, 2009, 02:30 PM

Quote:

Originally Posted by svarog $View Post$

My professor has the annoying habit of consistently giving us problems which are stated wrong (after few hours of

you eventually figure out it's not you that's making the mistake), but this time I think it happened on the midterm. Could someone please look at this:

For what s=? does $Var(X_n) ->_p0$ ?
$X_n$ is the sample mean of n independent $X_i$ (not identically distributed), where $X_i=i^s$ with $p=1/2$ and $X_i=-i^s$ with $p=1/2$ .

Here's my reasoning. $Var(X_i)=1/2*(i^{2s}+i^{2s})=i^{2s}$ so then $Var(X_n)=Var(1/n*sum(X_i))=1/n^2*sum(Var(X_i))=$ (for i from 1 to n)
$=1/n^2*sum(i^{2s})=O_p(n^{2s}/n^2)=O_p(n^{2(s-1)})$ as n->oo
This means the highest order of n in the sequence is 2(s-1). The sequence will go to 0 as n goes to infinity (for all terms) if the highest order also goes to 0. So 2(s-1)<0, thus s<1.
However, the problem asked me to prove that s<1/2. I asked the professor to make sure my assumptions are correct; he confirmed, however the result still disagrees. To be diplomatic, I wrote the solution until the last line (with $O_p$ ), however I wrote that the conclusion is s<1/2. (just to make sure i'm wrong no matter what happens

)) Opinions?

Opinion:
1) I guess you mean $Var(S_n)$ where $S_n=X_1+\cdots+X_n$ (or $\bar{X}_n$ )
2) $\sum_{i=1}^n i^{2s}\sim_{n\to\infty}\int_0^n x^{2s}dx=\frac{n^{2s+1}}{2s+1}$ , hence ${\rm Var}(\frac{S_n}{n})\sim_n \frac{n^{2s+1}}{(2s+1)n^2}=C n^{2s-1}$ ...