Local limit theorem

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  • #1
ehrenfest
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Homework Statement


This theorem is from Shiryaev. Can someone PLEASE explain how they are using the little o notation here. It makes no sense to me how they say that a FIXED real number is little o of something. I thought f(n) = o(g(n)) mean that for ever c>0 there exists an n_o such that when n => n_0, |f(n)|<c|g(n)|.

Homework Equations





The Attempt at a Solution

 

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Answers and Replies

  • #4
Hurkyl
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It makes no sense to me how they say that a FIXED real number is little o of something.
Where did they say that?
 
  • #5
ehrenfest
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Where did they say that?

It says for all [tex]x \in R^1[/tex]
 
  • #6
Hurkyl
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It says for all [tex]x \in R^1[/tex]
That doesn't sound like a FIXED real number to me. And besides, they clarify what they mean immediately afterwards.
 
Last edited:
  • #7
ehrenfest
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That doesn't sound like a FIXED real number to me.

They say for all [tex]x \in R^1[/tex] such that [tex]x = o(npq)^{1/6}[/tex]. To me that means that you can take any real number that you want and see if it is [tex]o(npq)^{1/6}[/tex]. I don't know how else to interpret it.
 
  • #8
Hurkyl
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I don't know how else to interpret it.
How about exactly how they interpreted it in the attachment you posted?
 
  • #9
ehrenfest
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How about exactly how they interpreted it in the attachment you posted?

That is precisely my question: How did they interpret it?
 
  • #10
Hurkyl
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How did they interpret it?
as [itex]n \rightarrow \infty[/itex],
[tex]
\sup_{\left\{ x : |x| \leq \psi(n) \right\}}
\left|
\frac{P_n(np + x \sqrt{npq})}{
\frac{1}{\sqrt{2\pi npq}} e^{-x^2 / 2}
} - 1
\right| \rightarrow 0,
[/tex]​
where [itex]\psi(n) = o(npq)^{1/6}[/itex].
 
  • #11
ehrenfest
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as [itex]n \rightarrow \infty[/itex],
[tex]
\sup_{\left\{ x : |x| \leq \psi(n) \right\}}
\left|
\frac{P_n(np + x \sqrt{npq})}{
\frac{1}{\sqrt{2\pi npq}} e^{-x^2 / 2}
} - 1
\right| \rightarrow 0,
[/tex]​
where [itex]\psi(n) = o(npq)^{1/6}[/itex].

OK. I will have to think about that. I am kind of confused about the sup above. Is \psi(n) fixed? If not, I don't understand how the sup is taken. Is it taken over all x AND over all \psi(n) that are [tex]o(npq)^{1/6}[/tex]?
 
  • #12
ehrenfest
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anyone?
 
  • #13
ehrenfest
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I FINALLY figured this out! Hurky! was right that they clarify what they mean-that just wasn't clicking for me.
 

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