Let $X_n$ a random variable of same law

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Discussion Overview

The discussion revolves around the behavior of a sequence of random variables \(X_n\) that share the same distribution, particularly focusing on the implications of the variance \(V(X_n)\) approaching zero as \(n\) increases. Participants are exploring how this condition relates to the convergence of the expected values \(E(X_n)\) and \(E(X_n^2)\) to a constant \(C\).

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to show that \(E(X_n) \longrightarrow C\) and \(E(X_n^2) \longrightarrow C^2\) given that \(V(X_n) \longrightarrow 0\).
  • Another participant suggests a method involving the transformation \(z = x - E(x)\) and proposes proving that \(E(z^2) = 0\) implies \(Z\) has a discrete probability density with \(P(z=0) = 1\).

Areas of Agreement / Disagreement

The discussion does not appear to have a consensus, as participants are exploring different approaches and methods without agreeing on a definitive solution or conclusion.

Contextual Notes

Participants have not provided specific assumptions or definitions that may be critical to the arguments, and the mathematical steps leading to the proposed conclusions remain unresolved.

Feynman
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:confused: Hello

I have a simple question :
Let $X_n$ a random variable of same law
If $V(X_n)\longrightarrow 0$ when $n\longrightarrow +\infty$
How schow that : $E(X_n)\longrightarrow C$ and $E(X_{n}^{2}\longrightarrow C^2$ and C is a constant?
Thanks
 
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Hello

[tex] <br /> I have a simple question :<br /> Let $X_n$ a random variable of same law<br /> If $V(X_n)\longrightarrow 0$ when $n\longrightarrow +\infty$<br /> How schow that : $E(X_n)\longrightarrow C$ and $E(X_{n}^{2}\longrightarrow C^2$ and C is a constant?<br /> Thanks[/tex]
 
Feynman, LaTex is down at the moment.
 
How far have you got with this one Feynman?

I suggest you let z = x - E(x) and then prove that E( z^2 ) = 0 implies that Z has a discrete probability denstiy with P(z=0) = 1.
 

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