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Math Help - Is my professor wrong again?

  1. #1
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    Is my professor wrong again?

    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?
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  2. #2
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    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}...
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