Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2210.16100

An OSSS-type inequality for uniformly drawn subsets of fixed size

Jacob van den Berg·Henk Don
View original source ↗Read source document ↗Export metadata ↓

Abstract

The OSSS inequality [O'Donnell, Saks, Schramm and Servedio, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), Pittsburgh (2005)] gives an upper bound for the variance of a function f of independent 0-1 valued random variables, in terms of the influences of these random variables and the computational complexity of a (randomised) algorithm for determining the value of f. Duminil-Copin, Raoufi and Tassion [Annals of Mathematics 189, 75-99 (2019)] obtained a generalization to monotonic measures and used it to prove new results for Potts models and random-cluster models. Their generalization of the OSSS inequality raises the question if there are still other measures for which a version of that inequality holds. We derive a version of the OSSS inequality for a family of measures that are far from monotonic, namely the k-out-of-n measures (these measures correspond with drawing k elements from a set of size n uniformly). We illustrate the inequality by studying the event that there is an occupied horizontal crossing of an R times R box on the triangular lattice in the site percolation model where exactly half of the vertices in the box are occupied.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jacob van den Berg, Henk Don. 2024-06-18. An OSSS-type inequality for uniformly drawn subsets of fixed size. https://arxiv.org/abs/2210.16100

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Distribution-uniform strong laws of large numbers

We revisit the question of whether the strong law of large numbers (SLLN) holds uniformly in a rich family of distributions, culminating in a distribution-uniform generalization of the Marcinkiewicz-Zygmund SLLN. These results can be viewed as extensions of Chung's distribution-uniform SLLN to random variables with uniformly integrable $q^\text{th}$ absolute central moments for $0 < q < 2$. Furthermore, we show that uniform integrability of the $q^\text{th}$ moment is both sufficient and necessary for the SLLN to hold uniformly at the Marcinkiewicz-Zygmund rate of $n^{1/q - 1}$. These proofs centrally rely on novel distribution-uniform analogues of some familiar almost sure convergence results including the Khintchine-Kolmogorov convergence theorem, Kolmogorov's three-series theorem, a stochastic generalization of Kronecker's lemma, and the Borel-Cantelli lemmas. We also consider the non-identically distributed case.

math.PR↗
arXiv2026

Malliavin Calculus for rough stochastic differential equations

In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and Lê (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a Hörmander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.

math.PR↗
arXiv2026

Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation

We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.

math.PR↗