How Can the Sum of Independent Normal Random Variables Be Represented?

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Discussion Overview

The discussion revolves around the representation of the sum of independent normal random variables, specifically the limit of the sum as the number of variables approaches infinity. Participants explore the implications of the central limit theorem (CLT) in this context and whether it applies without a normalizing factor.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions if the sum lim_{n\rightarrow \infty}\sum_{k=0}^n X_k can be represented by another random variable.
  • Another participant suggests considering the central limit theorem (CLT) in relation to the sum.
  • A different participant acknowledges the CLT but expresses uncertainty about its applicability due to the absence of a normalizing factor (1/n) in front of the sum.
  • One participant reiterates the original question about representing the sum and notes that the sum of n independent normal random variables has a normal distribution with variance n, but states that the distribution function does not converge as n approaches infinity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the applicability of the central limit theorem to the sum without a normalizing factor, indicating a disagreement or uncertainty regarding the interpretation of the theorem in this scenario.

Contextual Notes

There is an unresolved question regarding the conditions under which the central limit theorem applies, particularly in the absence of a normalizing factor. Additionally, the implications of the sum's variance as n increases are not fully explored.

Apteronotus
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For random variables Xk~N(0,1) is there any way of representing the following sum by another random variable?
[tex] lim_{n\rightarrow \infty}\sum_{k=0}^n X_k[/tex]

thanks
 
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What does the central limit theorem have to say about this question?
 
D H said:
What does the central limit theorem have to say about this question?

Thanks DH for your reply, but I've already taken the CLT into consideration. However, since there is no normalizing factor (1/n) in front of the sum I didn't think it applied.

Does it?
 
Apteronotus said:
For random variables Xk~N(0,1) is there any way of representing the following sum by another random variable?
[tex] lim_{n\rightarrow \infty}\sum_{k=0}^n X_k[/tex]

thanks
The random variable for sum to n is normal with a variance = n. The distribution function does not converge to anything as n -> ∞.
 

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