3
votes

### Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$

Such "concrete" formulas are addressed in Section 3.4. of my memoir Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory. This is in the ...

3
votes

### Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$

Given a collection of free random variables $X_i $, to evaluate a trace of the form
$$
\tau [ f_1 (X_{i_1 } ) f_2 (X_{i_2 } ) \ldots f_n (X_{i_n } ) ]
$$
where $i_{j+1} \neq i_{j} $, consider the ...

3
votes

Accepted

### 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 ...

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 ...

2
votes

Accepted

### 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,...

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 ...

1
vote

Accepted

### $n$-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 ...

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 ...

1
vote

### Perturbation results for statistical estimators

Robust statistics methodology by the likes of researchers like Rousseeuw, Tukey, Huber use such terminology in describing how much a sample can be perturbed by aberrant observations and outliers ...

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