I have a random vector $X = [X_1, X_2, \dots, X_n]^T$. I top-$k$ threshold it so that I get a new vector $Y = [Y_1, \dots, Y_n]$ where $Y_i = X_i$ if $X_i$ is in the top-k entries of $X$, and $Y_i=0$ otherwise. More succinctly, I select the top $k$ entries to stay the same, and set all other entries to 0.

What is the covariance matrix of the resulting vector? I want the quantity $cov(Y_i, Y_j)$ for all $i$, $j$. I'm especially interested in the spectrum of the covariance matrix, if possible.

We can assume the entries of $X$ are i.i.d. for simplicity. I would like to assume they are i.i.d. Gaussian, but maybe that will complicate things.

Current Progress

Not much, I'm afraid. I am aware of order statistics (https://en.wikipedia.org/wiki/Order_statistic), which would give me the distribution of the $k$th smallest value, but these distributions only seem characterized for a few specific distributions. Is there nothing that can be said for Gaussian random variables, or for more general cases?

I've found some related questions, but none exactly what I'm looking for.

  1. This looks at the min of a set of Gaussian rv
  2. Gives an approximation for the expected order statistic?
  3. Gives a formula for the expected order statistic, I imagine this will get messy fast...
  4. This gives an estimate of the variance of the kth order statistic?


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