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Convergence in Distribution

  1. Nov 5, 2009 #1
    1. The problem statement, all variables and given/known data

    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]

    2. Relevant equations



    3. 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?
     
  2. jcsd
  3. Nov 22, 2009 #2
    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?
     
  4. Nov 22, 2009 #3

    statdad

    User Avatar
    Homework Helper

    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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