# Convergence in Distribution

1. Nov 5, 2009

### cse63146

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

2. Relevant equations

3. 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?

2. Nov 22, 2009

### 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?

3. Nov 22, 2009

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