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Here is my computation. First of all, I had no idea whatsoever how to integrate implicit trigonometric functions, so I decided to switch to the integration with respect to the sides $x,y,z$ of the tri …

Not sure about the reference, but the "probability proof" is already nearly trivial. Just notice that you can run your assignment step by step, at each step either placing some particular weighted edg …

Warning: This is not a proper answer, just a dump of the thoughts I have had about this problem so far. Also, I'm not an expert in random matrix theory, so some bounds I'll be using may cry for improv …

I'll try to answer both questions at once. First, let's find the reason why the LHS is positive at all. I claim that the spectrum $\mu_1\le\mu_2\le\dots\le\mu_n$ of $2\sqrt{A^{1/2}BA^{1/2}}$ is domina …

I am not quite sure if the previous long discussion has already resulted in a full proof of anything but here is the crude bound that shows that the expectation in question is infinite. Let $X_i$ be i …

You can easily derive everything from the first principles by the following back of envelope computation: If we have a random walk starting anywhere, then after $t$ steps the expected number $EN(t)$ o …

You can prove more. Let $F(u,v)$ be any $1$-Lipschitz function on $[0,1]^2$ such that $F(u,v)<\min(u,v)$ inside the square. Then there exists a copula $D(u,v)$ such that $$ F(u,v)\le D(u,v)<\min(u,v) …

In the rewritten form it doesn't require any ingenuity at all. For brevity, introduce the notation $$ (Th)(s)=\frac 1\pi\int_0^s\frac{h(s-u)h(u)}{\sqrt{u(s-u)}}\,du=\frac 1\pi\int_0^1\frac{h(s(1-v))h( …

Certainly. All you need is $EX^2=+\infty$. Then the characteristic function $f_X(t)$ satisfies $\lim_{t\to 0}\frac{1-|f(t)|}{t^2}=+\infty$, so for every finite interval $I\subset \mathbb R$, we have $ …

Here is a positive answer to Q1. Step 1: Without loss of generality, each $X_i$ is symmetrically distributed. Indeed, consider an independent copy $X_i'$ of the family $X_i$ with the same joint distri …

I'm pretty sure that you figured it out by now, but I'll post it for the sake of completeness. Any unimodal $\sqrt{p}$ can be approximated by a sum $\sum c_k\chi_{I_k}$ with $I_1\subset I_2\subset\dot …

It seems like it is high time to handle this one. The main difficulty here is that there seems to be no conceptual reason for the inequality to be true: it just comes up valid before one numerical con …

I tried to implement my proposal in a C-code. That is a mixture of analytic and numeric integration. It does $10^6$ rectangles with half-percent relative precision in about 16 seconds, which is a bit …

Royen's proof of the Gaussian correlation conjecture (see here yields the following general statement: Let $(W_1(t),W_2(t))\in \mathbb R^{n_1+n_2}$ be a Gaussian vector for every fixed $t\in[0,1]$ wit …

Here is the uniqueness part. Suppose you have two functions $g_1$ and $g_2$ satisfying $g(x)-g^{-1}(x)=p(x)$ with the same $p$ (red and blue lines above the diagonal; the lines below the diagonal corr …