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Expectation of top-K selection of squared Gaussian random variables

Let $Z \sim \mathcal{N}(0,\sigma^2 I_n)$ be an $n$-dimensional Gaussian vector with independent $\mathcal{N}(0,\sigma^2)$ components. Consider the family of binary vectors in $\mathbb{R}^n$ with ...
Damien's user avatar
  • 146
1 vote

$n$-wise extension of covariance / correlation

If you denote random variables by capital $X\text{s},$ you should stick with that, and probably reserve the lower-case $x$ for arguments to its p.d.f. and c.d.f. and the like. There is the idea of the ...
Michael Hardy's user avatar
1 vote

$n$-wise extension of covariance / correlation

$\newcommand\Var{\operatorname{Var}}\newcommand\Cov{\operatorname{Cov}}$In conditions 1--3 in your post, the $x_i$'s are used improperly in place of the $X_i$'s. Also, the functions $C$ must of course ...
Iosif Pinelis's user avatar
2 votes

Points based partial ranking

This is an interesting question. An expert in social choice would have an interesting reply. As a semi-expert I can say semi-interesting things. As you may know, a common voting setting in social ...
usul's user avatar
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2 votes
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Relationship between fixed points and inversions in permutations

If we take $n=4,j=4$ then $k=1$ is possible (e.g. permutation $2431$). Looking at permutations with $j−1$ inversions we get $S_\beta=\{1432,2341,2413,3142,3214,4123\}$ and $S_\delta=\{2341,2413,3142,...
Peter Taylor's user avatar
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Expectation of the trace of inverse of a Gaussian random matrix

The following argument is quite similar to Carlo Beenakker's. For simplicity, I only consider the real case. I'm also going to use different symbols for the sizes of the matrices. Let $X$ be an $n \...
dohmatob's user avatar
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1 vote

Relationship between fixed points and inversions in permutations

A counterexample for $n = 4$ was given in the comments (2431, per @PeterTaylor). Monte Carlo simulation results of my own suggest counterexamples ought to increase with increasing $n$. Moreover, the ...
virtuolie's user avatar
  • 173
3 votes
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Evolution of the empirical mean of a list as we remove elements proportional to their value

A simple mean field calculation suggests to look at the solution to $$ \partial_t Q_t(k) = -\frac{k Q_t(k)}{\sum_\ell \ell Q_t(\ell)},\qquad Q_0 = P. $$ For all $n$, one would then expect a good ...
Martin Hairer's user avatar
1 vote

Evolution of the empirical mean of a list as we remove elements proportional to their value

I doubt this is an easy analysis, as the result is going to be affected by the distribution used for $P(k)$. I doubt there is a general form, but empirically there may be a well-behaved example. In ...
Henry's user avatar
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3 votes

Points based partial ranking

The question is in the area called social choice. You can look for textbooks that write on this topic, like Chapter 22 in "Game Theory" by Maschler, Solan, and Zamir. There are many ways to ...
Eilon's user avatar
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