Convergence in Distribution

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  • #1
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Homework Statement



Let Xn be a sequence of p dimensional random vectors. Show that

Xn converges in distribution to [tex]N_p(\mu,\Sigma)[/tex] iff [tex]a'X_n[/tex] converges in distribution to [tex]N_1(a' \mu, a' \Sigma a).[/tex]

Homework Equations





The Attempt at a Solution



[tex]E(e^{(a'X_n)t} = E(e^{(a't)X_n}) = e^{a't \mu + 0.5t^2(a' \Sigma a)}[/tex]

Hence, {a'Xn} converges [tex]N(a' \mu, a' \Sigma a).[/tex] in distribution.

Is that it?
 

Answers and Replies

  • #2
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hey, i can't seem to get this question.
what is the sum of: SIGMA (i=1 to n) of i(i+1)(i+2)... is the answer just infinity or is it some kind of weird expression that i have to find?
 
  • #3
statdad
Homework Helper
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pstar - you need to start your own thread - intruding into another's isn't appropriate.

To the OP:
You have the outline, but the rough edges need to be smoothed. For example, stating this equality

[tex]
E(e^{(a't)X_n}) = E^{a't\mu + 0.5t^2 (a' \Sigma a)}
[/tex]

isn't correct - there is a limit involved, correct?

The proof isn't long, and you've got the basic idea, but the details need to be included.
 

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