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For your first question, Rigollet has a series of notes(with minor typos) that dicusses basics of this kind of tail bounds. The result you mentioned in the "introduction" is actually the classic Hoeff …

I do not think the conclusion will hold for a general exchangeable sequence. In a more general case, you have to assume that U-statistics itself are not degenerated ($w_i\neq w_j$ for $i\neq j$) and t …

This is too long to be a comment. But I felt it serves as a pointer. Without loss of generality we assume $X_{1},X_{2},X_{3}$ share the same image domain $[0,1]$. For a specific $\pi\in S_{3}$, the …

No, they just used sloppy notations. They omitted what we called "attainment argument". Suppose the attainment of supremum occurs at $h_{0}\in\mathcal{H}$,$$Pr\left[\sup_{h\in\mathcal{H}}\left|\wideh …

$$\mathbb{P}(X-Y<t\mid X>Y) =\frac{\mathbb{P}(X-Y<t,X>Y)}{\mathbb{P}(X>Y)} =\mathbb{P}(X-Y<t)$$ since $X(\omega)>Y(\omega),\ \forall\omega\in\Omega $. $$\mathbb{P}(X-Y<t)=\int_{0}^{\infty}dy\int_{0}^ …

The problem can be solved if the distribution $f(\cdot)$ is in a Levy stable distribution family. In your concrete example, since the $d\dim$ normal distribution $N(\mu,I_d)$ is "additive", the exact …

No. Vector space structure is not enough, we actually need a compatible lattice structure to make things work. To apply the conditional expectation operator $E(\bullet\mid Y)$ onto the Hilbert space c …

Iosif Pinelis provided a very nice answer, however, I would like to provide a more comprehensive answer to this question. I think the title is a bit misleading because we do not actually need the orde …

I think it is just called (scaled) supremum $L_1$ norm and it is mostly studied in Bayesian nonparametric estimation literature, especially posterior consistency. The following paper investigate condi …

The quote actually belongs to C.G.Rota. Because cumulant sequences are closed under addition while moment sequences are not. That makes cumulant a more tractable algebraic structure altogether. Altho …

This is the more from decision theory, which could also be regarded as part of statistics if you regard the procedure of making a statistical decision as sort of de-randomization from a categorical vi …

According to your motivation, Numerical computation of $\rho$ is notoriously inaccurate and problematic, and so KL divergence or even $L^1$ distance are problematic as well. I obtain $f$ anyway, …

This $y = \frac{<x_1, x_2>}{\|x_1\|\|x_2\|}$ is the distribution of the $cos \theta$ where $\theta$ is known as the canonical angle/principal angle of two random vectors $x_1,x_2$. $cos\theta$ is know …

$g_\theta(x):=e^{-(\theta x-1)^2/2}=\exp(-\frac{1}{2\frac{1}{\theta^2}}[x-\frac{1}{\theta}]^2)$ is the kernel of $N(\frac{1}{\theta},\frac{1}{\theta^2})$ with density $\frac{1}{\sqrt {2\pi \frac{1}{\t …

Problem 1 For a general closed form expression of the joint density of order statistics, it is known intractable: A problem that is closely related to the first-passage time problem is that o …