Suppose we have n variables which all have pairwise correlation of 1, and we’re further given that the variance of the sum of the variables is 1. What is the lower bound we can put on the sum of the individual variances?
I know this corresponds to minimising the trace of the covariance matrix given a fixed sum of all its elements, but I’m not sure the best steps to take to do this given the diagonal elements aren’t independent of the off-diagonal elements
Any help would be appreciated!
Let the given random variables be $a_1X+b_1,a_2X+b_2,a_3X+b_3,\dots,a_nX+b_n$, where $a_i$ are positive constants and $b_i$ are real constants.
So the variance of sum of the random variables is $\left(\sum\limits_{i=1}^n a_i\right)^2\mathrm{Var}(X)=1$. The sum of the individual variances is $\left(\sum\limits_{i=1}^na_i^2\right)\mathrm{Var}(X)\ge \frac{\left(\sum\limits_{i=1}^n a_i\right)^2}{n}\mathrm{Var}(X)=\frac1n$ (By Cauchy-Schwarz inequality).
This is the required lower bound.