A short proof and a generalization of the BKR-inequality

Grünbaum不等式在RCD$(0,N)$-空间中的推广

A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces

将Grünbaum不等式推广至非负Ricci曲率空间（RCD(0,N)空间及加权黎曼流形），利用等距分裂定理和Busemann函数给出体积下界，并推广至非均匀分布，建立刚性结果并研究稳定性。

Generalization and refinement of Khintchin's inequality

We derive the exponential as well as power decreasing tail estimations for normed sums of centered independent identical distributed (or not) random variables on the Khintchine's form. We consider arbitrary, in particular, non-Rademacher's variables and not only Lebesgue-Riesz rearrangement invariant norms for the random variables. We intend to calculate the value of correspondent limit.

Violation of contextual generalization of the Leggett-Garg inequality for recognition of ambiguous figures

We interpret the Leggett-Garg (LG) inequality as a kind of contextual probabilistic inequality in which one combines data collected in experiments performed for three different contexts. In the original version of the inequality these contexts have the temporal nature and they are given by three pairs of instances of time, $(t_1, t_2), (t_2, t_3), (t_3, t_4),$ where $t_1 < t_2 < t_3.$ We generalize LG conditions of macroscopic realism and noninvasive measurability in the general contextual framework. Our formulation is done in the purely probabilistic terms: existence of the context independen…

Opial 不等式的尖锐统一推广

Sharp unifying generalizations of Opial's inequality

本文提出基于分布函数的 Opial 不等式尖锐统一推广，涵盖连续与离散形式及任意阶导数，改进了已有常数，并讨论加权形式与相关 Wirtinger 不等式。

On some generalization of Lorden's inequality for renewal processes

In queueing theory, Lorden's inequality can be used for bounds estimation of the moments of backward and forward renewal times. Two random variables called backwards renewal time and forward renewal time for this process are defined. Lorden's inequality it's true for the renewal process, so expectations of backward and forward renewal times are bounded by the relation of expectation of moment of the random variable for any fixed moment of time, where random variables are i.i.d. We generalised and proved a similar result for dependent random variables with finite expectations, some constant C a…

Chebyshev's inequality for Banach-space-valued random elements

In this paper, we obtain a new generalization of Chebyshev's inequality for random elements taking values in a separate Banach space.

Poincaré Inequality for Dirichlet Distributions and Infinite-Dimensional Generalizations

For any $N\ge 2$ and $å:=(å_1,\cdots, å_{N+1})\in (0,\infty)^{N+1}$, let $μ^{(N)}_å$ be the corresponding Dirichlet distribution on $\DD:= \big\{ x=(x_i)_{1\le i\le N}\in [0,1]^N:\ \sum_{1\le i\le N} x_i\le 1\big\}.$ We prove the Poincaré inequality $$μ^{(N)}_å(f^2)\le \ff 1 {å_{N+1}} \int_{\DD}\Big\{\Big(1-\sum_{1\le i\le N} x_i\Big) \sum_{n=1}^N x_n(\pp_n f)^2\Big\}μ^{(N)}_å(\d x)+μ^{(N)}_å(f)^2,\ f\in C^1(\DD)$$ and show that the constant $\ff 1 {å_{N+1}}$ is sharp. Consequently, the associated diffusion process on $\DD$ converges to $μ^{(N)}_å$ in $L^2(μ^{(N)}_å)$ at the exponentially rate…

The GHS inequality for the Potts Model

In this note we are concerned about the generalization of the GHS inequality for the Potts model. We also obtain by a different method the proof of the GHS inequality for the Ising model. We take advantage of a polynomial expansion and we derive that some of the variables of the polynomial can be separated

Generalization Bounds for Domain Adaptation

In this paper, we provide a new framework to obtain the generalization bounds of the learning process for domain adaptation, and then apply the derived bounds to analyze the asymptotical convergence of the learning process. Without loss of generality, we consider two kinds of representative domain adaptation: one is with multiple sources and the other is combining source and target data. In particular, we use the integral probability metric to measure the difference between two domains. For either kind of domain adaptation, we develop a related Hoeffding-type deviation inequality and a symmetr…

$P$-分拆的多元相关不等式

Multivariate correlation inequalities for $P$-partitions

本文将 Fishburn、Daykin–Daykin–Paterson 及 Brightwell–Felsner–Trotter 等偏序集相关不等式推广到多元 $P$-分拆情形，证明依赖 Ahlswede–Daykin 不等式的多元推广。

Fractional generalizations of Young and Brunn-Minkowski inequalities

A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. It is shown that the generalized Brunn-Minkowski conjecture is true for convex sets; an application of this to the law of large numbers for random sets is described.