1
$\begingroup$

What are the conditions under which a distribution reduces down a dimension?

For example, suppose I have a 2D gaussian distribution for X and Y. Under what condition(s) on Y does the distribution "reduce" down to a 1D gaussian distribution for X?

I don't want to say more so as not to bias the answers.

  • 0
    @Henry Isn't the _only_ bijective relationship between $X$ and $Y$ such that both $X$ and $Y$ are Gaussian$a$linear (or affine) relationship: $Y = aX+b$ for real numbers $a$ and $b$ with $a$ being allowed to be $0$ if we are amenable to calling a constant a Gaussian random variable with zero variance?2011-10-30

2 Answers 2

1

Given your assumptions, the variance of Y should be much less than the variance of X. More generally, when the "principal component" is neither X nor Y but a combination of the two, the solution is given by Principal component analysis. More generally still, see Dimension reduction.

0

I don't think there are any such conditions, unless $\Sigma$ is allowed to be non-negative definite (and not strictly positive definite). If either of the marginal variances approaches 0 or the correlation between the two approaches 1, the distribution effectively collapses to 1 dimension and you still have a non-negative definite $\Sigma$. However, I don't think this is legal because $\Sigma$ would be singular and the joint density would be undefined.