Convergence in Distribution for Random Vectors

[cse63146]

Homework Statement

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

Xn converges in distribution to $$N_p(\mu,\Sigma)$$ iff $$a'X_n$$ converges in distribution to $$N_1(a' \mu, a' \Sigma a).$$

Homework Equations

The Attempt at a Solution

$$E(e^{(a'X_n)t} = E(e^{(a't)X_n}) = e^{a't \mu + 0.5t^2(a' \Sigma a)}$$

Hence, {a'Xn} converges $$N(a' \mu, a' \Sigma a).$$ in distribution.

Is that it?

 

[pstar]

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?

 

[statdad]

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

$$E(e^{(a't)X_n}) = E^{a't\mu + 0.5t^2 (a' \Sigma a)}$$

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.