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Quantum theory allows for correlations between the outcomes of distant measurements that are inconsistent with any locally causal theory, as demonstrated by the violation of a Bell inequality. Typical demonstrations of these correlations…

Fix a graph $H$ and some $p\in (0,1)$, and let $X_H$ be the number of copies of $H$ in a random graph $G(n,p)$. Random variables of this form have been intensively studied since the foundational work of Erd\H{o}s and R\'{e}nyi. There has…

Hanson-Wright inequality provides a powerful tool for bounding the norm $|\xi|$ of a centered stochastic vector $\xi$ with sub-gaussian behavior. This paper extends the bounds to the case when $\xi$ only has bounded exponential moments of…

Let $(X_1,\dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots,S_m)$ be a free semicircular…

Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be written: \[ \int h_A^\beta d\mu \ge 2 \mu(A)(1-\mu(A)). \] Here $\mu$ is uniform measure on $V=\{0,1\}^n$ ($=V(Q_n)$); $\beta=\log_2(3/2)$; and, for $S\subseteq V$…

We prove the sharp isoperimetric inequality $$ \mathbb{E} \,h_{A}^{\log_{2}(3/2)} \geq \mu(A)^{*} (\log_{2}(1/\mu(A)^{*}))^{\log_{2}(3/2)} $$ for all sets $A \subseteq \{0,1\}^n$, where $\mu$ denotes the uniform probability measure,…

For a random variable $N = 0, 1, 2, \ldots$ we study the following question: When does the sum of $N$ many independent and identically distributed copies of a random variable $X$ have the same law a a nontrivial rescaling of $X$? We show…

Let $\{X_n,n\ge1\}$ be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call $X_n$ a $\delta$-record if $X_n>\max\{X_1,...,X_{n-1}\}+\delta$, where $\delta$ is an integer…

Consider the random quadratic form $T_n=\sum_{1 \leq u < v \leq n} a_{uv} X_u X_v$, where $((a_{uv}))_{1 \leq u, v \leq n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal, and $X_1,$ $X_2, \ldots, X_n$ are i.i.d.…

Let $\mathbf{X}(n) \in \mathbb{R}^d$ be a sequence of random vectors, where $n\in\mathbb{N}$ and $d = d(n)$. Under certain weakly dependence conditions, we prove that the distribution of the maximal component of $\mathbf{X}$ and the…

The notorious quantum measurement problem brings out the difficulty to reconcile two quantum postulates: the unitary evolution of closed quantum systems and the wave-function collapse after a measurement. This problematics is particularly…

The sample correlation coefficient $R$ plays an important role in many statistical analyses. We study the moments of $R$ under the bivariate Gaussian model assumption, provide a novel approximation for its finite sample mean and connect it…

Quantum processes can exhibit scenarios beyond a fixed order of events. We propose information inequalities that, when violated, constitute sufficient conditions to certify quantum processes without a fixed causal order -- causally…

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with…

Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and…

The aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For a fixed probability measure $q^n$ on $\mathcal X^n$, ($\mathcal X$ is a finite set),…

In this paper, we compare two variances of maxima of $N$ standard Gaussian random variables. One is a sequence of $N$ i.i.d. standard Gaussians, and the other one is $N$ standard Gaussians with covariances $\sigma_{1,2}=\rho \in(0,1)$ and…

Let (X_n,Y_n) be i.i.d. random vectors. Let W(x) be the partial sum of Y_n just before that of X_n exceeds x>0. Motivated by stochastic models for neural activity, uniform convergence of the form $\sup_{c\in I}|a(c,x)\operatorname…

Composite endpoints are widely used in cardiovascular clinical trials to improve statistical efficiency while preserving clinical relevance. The Win Ratio (WR) measure and more general frameworks of Win Statistics have emerged as…

Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ taking the steps $(1,0)$, $(-1,1)$ and $(0,-1)$ with probabilities $\lambda < (\mu_1\neq \mu_2)$; in particular, $X$ is assumed stable. Let $\tau_n$ be the first time $X$ hits…