0
$\begingroup$

So, I have a set of dependent Gaussian RVs $\{X_k\}_{k=1}^{N}$ with known joint PDF (zero mean and given covariance matrix). I'm interested in whether we can compute the quantity: $$ \mathbb{E}\left[\left(\sum_{k=1}^{N}{X_k^2}\right)^{\beta}\right], $$ where $\beta > 0$.

I tried thinking about the sum and see if I can linked to Chi-squared distribution, but the problem is that the set of variables in question are dependent.

$\endgroup$
  • $\begingroup$ math.stackexchange.com/questions/442472/… This should be related. $\endgroup$ – BGM Sep 10 '18 at 5:37
  • $\begingroup$ Thanks for directing me to that link. The thing is that the power here is making the problem harder or at least this is what I'm thinking. $\endgroup$ – Jeremy Sep 10 '18 at 5:43
0
$\begingroup$

Is $\beta$ an integer or what?

If what you actually know is that the vector $\vec X=(X_1,\ldots,X_N)$ follows a multivariate normal distribution with zero mean and covariance matrix $\Sigma$, then you have joint density $$f_{\vec X}(\vec x)=\frac1{\sqrt{2\pi \det(\Sigma)}}e^{-\frac12\vec x^T\Sigma^{-1}\vec x},$$ where $\vec x=(x_1,\ldots,x_N)^T$.

So, as a very general answer, we can say that $$E\left[\left(\sum_{k=1}^N X_k^2\right)^\beta\right]=\int_{\mathbb R^N}\left(\sum_{k=1}^N x_k^2\right)^\beta \cdot f_{\vec X}(\vec x)\quad dx_1\ldots dx_N,$$ but I can't think of a general expression for this integral.

$\endgroup$
  • $\begingroup$ $\beta$ is a real number. I tried writing down what you have written by I fail to see how can I possibly move forward. Would it change something if $\beta$ was an integer? using binomial formula? $\endgroup$ – Jeremy Sep 10 '18 at 6:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.