Maciej Skorski
This paper develops sharp bounds on moments of sums of k-wise independent bounded random variables, under constrained average variance. The result closes the problem addressed in part in the previous works of Schmidt et al. and Bellare, Rompel. We The work also discuss discusses other applications of independent interests, such as asymptotically sharp bounds on binomial moments.
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This paper describes the construction of a lower bound for the tails of general random variables, using solely knowledge of their moment generating function. The tilting procedure used allows for the construction of lower bounds that are tighter and more broadly applicable than existing tail approximations.
The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables. In all three cases, the bounds are tight in the sense that there exists easy examples where the inequalities become equality. Here we will show that through a simple smoothing using auxiliary randomness, that each of the three bounds can be cut in half. In many common cases, the halving can be achieved without the need for the auxiliary randomness.
We derive simple but nearly tight upper and lower bounds for the binomial lower tail probability (with straightforward generalization to the upper tail probability) that apply to the whole parameter regime. These bounds are easy to compute and are tight within a constant factor of $89/44$. Moreover, they are asymptotically tight in the regimes of large deviation and moderate deviation. By virtue of a surprising connection with Ramanujan's equation, we also provide strong evidences suggesting that the lower bound is tight within a factor of $1.26434$. It may even be regarded as the natural lowe…
Chernoff bounds for branching random walks
本文提出了一种更一般的分支随机游走框架，并由此推导出BRW的Chernoff型界，同时讨论了随机游走与BRW之间的关系。
Chernoff Bounds for Tensor Expanders on Riemannian Manifolds Using Graph Laplacian Approximation
本文提出一种基于图拉普拉斯近似的张量 Chernoff 界方法，用于分析黎曼流形上随机游走的尾概率，通过谱相似性确保近似精度，扩展了高维随机对象的尾界理论。
For a wide class of monotonic functions $f$, we develop a Chernoff-style concentration inequality for quadratic forms $Q_f \sim \sum\limits_{i=1}^n f(η_i) (Z_i + δ_i)^2$, where $Z_i \sim N(0,1)$. The inequality is expressed in terms of traces that are rapid to compute, making it useful for bounding p-values in high-dimensional screening applications. The bounds we obtain are significantly tighter than those that have been previously developed, which we illustrate with numerical examples.
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…
We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $μ$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mix…
In this note, we prove a tight lower bound on the joint entropy of $n$ unbiased Bernoulli random variables which are $n/2$-wise independent. For general $k$-wise independence, we give new lower bounds by adapting Navon and Samorodnitsky's Fourier proof of the `LP bound' on error correcting codes. This counts as partial progress on a problem asked by Gavinsky and Pudlák.
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（$t\ge2$）。
Approximate independence of permutation mixtures
本文提出控制χ²散度的新方法，证明了高维可交换混合分布与其独立同分布对应物的统计距离界限，并得出新的de Finetti风格定理及差分隐私等统计结果。
Studying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter $λ$ tending to infinity. Finding the independence ratio of random $d$-regular graphs for some fixed degree $d$ has received much attention both in random graph theory and in statistical physics. For $d \geq 20$ the problem is conjectured to exhibit 1-step replica symmetry breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio was confirmed for (very) large $d$ in a breakthrough paper by Ding, Sly, and Sun. Furthermore, the so-called interpolation method s…
We present novel and sharp lower bounds for higher load moments in the classical problem of mapping $M$ balls into $N$ bins by $q$-universal hashing, specialized to the case when $M=N$. As a corollary we prove a tight counterpart for the result about min-entropy condensers due to Dodis, Pietrzak and Wichs (CRYPTO'14), which has found important applications in key derivation. It states that condensing $k$ bits of min-entropy into a $k$-bit string $ε$-close to almost full min-entropy (precisely $ k-\log\log(1/ε)$ bits of entropy) can be achieved by the use of $q$-independent hashing with $q= \lo…
Decoupling Maximal Inequalities
本文研究极大不等式在非独立随机变量下的估计问题，证明对非负随机变量而言，两两独立足以使极大不等式表现与独立版本相当，且条件可放宽至可量化违反的负相关情形。
We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of $K_{d,d}$ maximizes the number of independent sets and the independence polynomial of a d-regular graph. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of $K_{d,d}$. Using this we prove the asymptotic upper matching conjecture of Friedlan…
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