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As suggested by my coauthor, and proved in the same paper linked in the question, the conjectured upper bound is indeed true. I reproduce the proof here for completeness: Let $p = TV(x, y)$. We want t …

Let $\Delta_n$ be the set of all probability vectors on $n$ points, also known as the $n$-simplex. Let $x, y \in \Delta_n$ be two probability vectors, that is, $\sum_{i=1}^n x_i = 1$ and $x_i \geq 0$, …

This has been solved in a joint paper with a colleague that is forthcoming. The key is to consider an interpolating probability measure $\rho$ of $\mu$ and $\nu$ defined by $\rho_i = \mu_i \nu_i / \su …

T. Royen proved the Gaussian correlation inequality in the context of Gamma distributions back in 2014, which was since popularized by Latala and Matlak. The properties of Gaussian integration seem he …

Assuming $Y_{2i+1} + Y_{2i+2}$ is independent of $Y_{2i+3} + Y_{2i + 4}$, then the answer to this question answers yours in the affirmative. To prevent non-degeneracy, your condition that "distributio …

I am looking for any remotely related reference for the following problem, for which I have not the least clue what techniques would be useful. Consider a discrete probability space $\Omega = \{x, y, …

Cross-posting from math stack-exchange since it's not getting any visibility there. I am given a function $F: \{[0, y]: y \in I\} \to \Sigma(I)$, such that $\lambda(F([0, y])) = y$, and $F([0, y]) \s …

Besides some trivial lower bound because of the range constraint, there is not much you can say here. I can let $Y$ be $X$ with probability $> k$ for each $X$ value and anything else otherwise. There …

I believe the conjecture is true for sufficiently small $C$. Previously I was trying to disprove it using Large deviation theory. But I missed a sign at the last step. But the same argument can be tur …

Inspired by Nate's answer, here is why it cannot be differentiable. First for each $y \in [0,1]$, there must be some $x \in [0,1]$ such that $h(x,y) = 0$, otherwise by compactness of the unit interval …

Unless there is something I didn't understand about your Brownian motion statement, the result is not true in general. Consider even a deterministic example, where $X_t^n = \frac{1}{2n} 1_{|t| < n}$. …

I am looking for a finitely parameterized family of non-atomic distributions $D(\vec{\lambda})$, $\vec{\lambda} \in \mathbb{R}^k$ for some finite $k$, such that if $X \sim D(\vec{\lambda}_1)$ and $Y \ …

$X/|X|$ is almost surely a uniformly random element on the unit sphere of dimension $n-1$; this is not the same thing as a rotation, which is a matrix. Since a multivariate standard Gaussian vector is …

Suppose I have a set of random vectors $f(a_1, \ldots, a_\ell) := (v_1, \ldots, v_m) \subset F_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a_i$'s are independent, u …

For the root system $A_n$, taking the limit $q = t^\alpha$ and $t \to 1$, and letting $Y = (t-1) X -1$ one obtains from the Macdonald operator the so-called Sekiguchi-Debiard operator: $$D_\alpha(X) …