On notions of Q-independence and Q-identical distributiveness

成对和三重独立同分布随机变量的经典中心极限定理反例的完善

A completion of counterexamples to the classical central limit theorem for pair- and triplewise independent and identically distributed random variables

本文完善了经典中心极限定理的反例研究，证明对于任意具有有限方差且不集中于单点的概率分布，均存在三重独立同分布随机变量序列使其标准化部分和依分布收敛于非正态分布，并可推广至 $ k $ 重独立性。

Hidden independence in unstructured probabilistic models

We describe a novel way to represent the probability distribution of a random binary string as a mixture having a maximally weighted component associated with independent (though not necessarily identically distributed) Bernoulli characters. We refer to this as the latent independent weight of the probabilistic source producing the string, and derive a combinatorial algorithm to compute it. The decomposition we propose may serve as an alternative to the Boolean paradigm of hypothesis testing, or to assess the fraction of uncorrupted samples originating from a source with independent marginals.…

A Characterization of the Normal Distribution by the Independence of a Pair of Random Vectors

Kagan and Shalaevski 1967 have shown that if the random variables $X_1,\dots,X_n$ are independent and identically distributed and the distribution of $\sum_{i=1}^n(X_i+a_i)^2$ $a_i\in \mathbb{R}$ depends only on $\sum_{i=1}^na_i^2$ , then each $X_i$ follows the normal distribution $N(0, σ)$. Cook 1971 generalized this result replacing independence of all $X_i$ by the independence of $(X_1,\dots, X_m) \textrm{ and } (X_{m+1},\dots,X_n )$ and removing the requirement that $X_i$ have the same distribution. In this paper, we will give other characterizations of the normal distribution which are fo…

线性趋势随机变量序列中记录事件间的相关性

Correlations between record events in sequences of random variables with a linear trend

独立同分布随机变量序列中记录的统计是一个经典的研究课题。最早的结果之一是关于记录事件的随机独立性。最近，记录的统计数据超出了i.i.d.的情况。随机变量受到了很多关注，但记录事件的独立性问题还没有得到系统的解决。在本文中，我们研究了这个问题的情况下，独立的，不同分布的随机变量，特别是线性移动的平均值的随机变量。我们发现了一个丰富的模式的积极和消极的相关性，并显示他们的渐近性是如何确定的普遍性类的极值统计。

受限独立性下依赖维度相关间隙界

Dimension Dependent Correlation Gap Bounds under Restricted Independence

该研究解决了成对独立相关间隙的$n=4$界及最坏情形紧界问题，证明$n=4$时$4/3$界普遍成立且紧，最坏情形下成对独立相关间隙渐近达$e/(e-1)$，并推广到$t$阶独立（$t\ge2$）。

Tight Probability Bounds with Pairwise Independence

While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results in this direction. Firstly, when $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided in closed-form for any input marginal probability vector $\mathbf{p} \in [0,1]^n$. To prove the result, we show the existence of a positively correlated Bernoulli random vector with transformed bivariate pr…