Characteristic Function of a Multivariate Cauchy

Hello, I am very lost with the following problem, is there a neat way of doing it, or just solving integral after integral is the way to go?

Any hint would be greatly appreciated.

Let $\displaystyle X \sim N_p(0,\Sigma)$ be independent of $\displaystyle Z \sim N(0,1)$. Show that $\displaystyle Y=X/Z$ has characteristic function

$\displaystyle \varphi_Y(t) = \exp\{-(t^T \Sigma t)^{1/2}\}$

Thanks in advance,

Re: Characteristic Function of a Multivariate Cauchy

Hey akolman.

Recall that the Moment Generating Function of a variable Y is E[e^(tY)] and in this case Y=X/Z and the expectation of E[X/Z] = Integral (over whole of R^2) (x/z)*f(x,z)dxdz.

The characteristic function of a random variable is simply MGF_Y(it) where MGF_Y(t) = E[e^(tY)] so the characteristic function is simply E[e^(itY)].

To start, setup your integral and see how you go.