Papers

Non-exchangeability and radial asymmetry have got substantial attention in the literature, but relation between the two has not been studied yet. We propose the question: if a copula has a given degree of radial asymmetry, what is the range of possible values of the measure of non-exhangeability. We determine the set of all points whose first coordinate is the radial asymmetry of a copula and the second coordinate is its non-exchangeability. We show that this set is a non-convex hexagon.

In this paper we construct a one-parameter family of periodic Airy line ensembles $\mathcal{A}^{\mathrm{per},R}$ for $R>0$ as weak scaling limits of Borodin's periodic Schur processes. We establish several properties of $\mathcal{A}^{\mathrm{per},R}$, including a determinantal point process structure with an explicit correlation kernel. We also show that, after a parabolic shift and normalization, $\mathcal{A}^{\mathrm{per},R}$ satisfies, on its domain $[-R/2,R/2]$, the same Brownian Gibbs property enjoyed by the usual Airy line ensemble. A main technical contribution of the paper is a general tightness criterion for determinantal line ensembles, which extends the argument used by Johansson in [Comm. Math. Phys. 242 (2003), 277-329] to prove a functional limit theorem for the discrete polynuclear growth model.

Closed-form formulas are derived for the cumulative distribution function and odd-order absolute moments of the variance-gamma distribution. As a special case, closed-form expressions are deduced for the cumulative distribution function and odd-order absolute central moments of the product of two correlated zero-mean normal random variables.

This is preliminary, AI-generated note on the Dudley-type bound for stationary Bernoulli processes postulated by M. Talagrand based on the unpublished notes of the authors and the recent note \cite{invariant} which in turn is an alternative proof of the result by Hu, Wang, Wu.

We investigate when a Poincaré inequality forces the underlying measure to be doubling, with particular attention to the space \(X\) controlling the oscillation. We identify \(L\log L\) as the sharp threshold: under a natural local growth condition on the measure, doubling is forced precisely when \(L\log L(0,1)\not\hookrightarrow X\). We also obtain a doubling criterion under a weaker geometric assumption when the oscillation norm is sufficiently stronger than \(L\log L\). We show that this threshold is sharp in every fixed Euclidean dimension, using a dimension-free local \(L\log L\)-Poincaré inequality for log-concave measures and non-doubling Gaussian examples. We then turn to infinite-dimensional spaces, where doubling is impossible for the measures under consideration. Nevertheless, the \(L\log L\)-Poincaré inequality continues to hold, and we establish sharp characterizations for infinite product models, Gaussian measures on Hilbert spaces under power-type small-ball estimates, and Wiener measure with the uniform norm.

Spectral duality structures (SDS) explain why a distinguished family of initial distributions makes the absorption probabilities of resetting processes insensitive to the resetting mechanism. Here we study their axioms. We prove that the six defining conditions are mutually independent and determine what each one contributes. The key tool is a structure theorem valid for an arbitrary permutation and an arbitrary positive weight: every insensitive distribution is supported on a union of cycles balanced at a common level, and the level of a cycle is fixed by the geometric mean of the weight on it. From it, positivity selects the admissible root, the constant orbit product is equivalent to the existence of an insensitive distribution of full support, and the involution is equivalent to the response rank being equal to the number of free weight parameters. Without the duality relation, insensitivity of a single distribution is cheap and generically isolated; duality turns it into structural neutrality of the separatrix. The full-rank condition splits into two layers: a weaker condition is equivalent to maximal response rank, while full rank also excludes insensitivity at any other level. We then show that a single SDS is internally rigid: compatible semigroup structures force a constant weight, compatible abelian group structures obey a precise arithmetic condition, congruences form a simple lattice, and the automorphism group is computed explicitly. Finally, the axioms do not determine the orientation of the response; in the two-site case we give the exact condition under which the geometric and analytic sign criteria agree.

This paper deals with the pullback measure attractor for a non-autonomous distribution-dependent stochastic lattice wave equation with nonlinear noise on $\mathbb{Z}$. We establish well-posedness in the energy space $\ell^{2}\times\ell^{2}$ and show that the solution generates a continuous non-autonomous dynamical system on $\mathcal{P}_{4}(\ell^{2}\times\ell^{2})$, which possesses a unique pullback measure attractor $\mathcal{A}$. Furthermore, we prove the convergence of the attractor $\mathcal{A}^{\varepsilon}$ as $\varepsilon\to 0^{+}$ to the attractor of the corresponding distribution-independent limit system.

This paper investigates the large deviation principle for 2D stochastic $g$-Navier–Stokes equations driven by multiplicative noise and defined on an unbounded channel-like domain. We first prove the existence and uniqueness of probabilistically strong solutions for the stochastic equation under small noise perturbations by means of the local monotonicity method. Subsequently, we show the well-posedness of solutions to the corresponding control equation and the convergence in distribution of the solutions for the stochastic equation as the noise intensity tends to zero. Finally, we establish the Freidlin-Wentzell type large deviation principle for the solutions of the stochastic equation by using the weak convergence method. To overcome the lack of compactness of Sobolev embedding on unbounded domains, we employ a scalar stream function together with a truncation technique to derive uniform tail estimates for the solutions of the control equation.

Motivated by a question of Hochman, we investigate whether two deterministic time measures or time sets can be distinguished almost surely from a single planar Brownian image. Using rescaled correlation integrals, we give a criterion for mutual singularity of the laws of the image measures, even when the time measures have the same dimension. For self-similar time sets satisfying the open set condition, we also recover the natural occupation measure from the compact image. Together, these results allow us to distinguish lattice and non-lattice self-similar sets and their natural measures within this class. For a naturally parametrised family with two contractions, with fixed dimension, each parameter can be distinguished from all but finitely many others. If the dimension is sufficiently small, every distinct parameter can be distinguished. The results are formalised in Lean.

We study Fisher information about the amplitude of a known smooth mean shift from the first Dirichlet eigenvalue and spatial responses of a Liouville random medium. The complete response record determines the speed measure and the Gaussian free field, whereas every finite record strictly loses information; an optimal design exists in each finite-dimensional probe space. We quantify this gap as the known Liouville parameter $\gamma$ tends to zero. If the probe space contains an explicit matching probe, the minimum loss equals $\gamma^2$ times a positive compression error for a deterministic second-chaos operator, up to $o(\gamma^2)$, with designs allowed to depend on $\gamma$. After enlarging the probe dictionary, the optimal $m$-response coefficient is asymptotic to $\|Lg\|_{L^1(D)}^2/(8m)$, where $L=-(2\pi)^{-1}\Delta_D$ on the unit disk $D$. A quadratic correction of the readouts attains the same fixed-design loss to second order.

In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all $\alpha$-$z$ relative entropies for $\alpha<1$. This demonstrates the first operational interpretation of these relative entropies.

We study the limiting distribution of the largest real part of the eigenvalues of truncated unitary matrices. Let \(U_m\) be an \(m\times m\) Haar-distributed unitary matrix, and let \(Z_1,\ldots,Z_n\) be the eigenvalues of its upper-left \(n\times n\) principal submatrix. We determine the limiting distribution of \(\max_{1\le i\le n}\Re Z_i\) under a unified normalization: the limiting law is Gumbel when $q:=m-n\to\infty$ and is of Weibull type when $q$ is fixed. The fixed-\(q\) limiting laws converge to the Gumbel law as \(q\to\infty\). We also give a unified formulation of the normalizing constants for the spectral radius that recovers the previously known limiting laws.

We resolve a conjecture of Łuczak, Mieczkowska and Šileikis on the maximal tail probability of sums of independent and identically distributed nonnegative random variables. Specifically, for every integer $k\ge1$ and every $0\le x\le1/k$, we prove \[ \sup_{\mathbb E X_1\le x}\mathbb{P}(X_1+\cdots+X_k\ge1) =\max\{1-(1-x)^k,(kx)^k\}, \] where the supremum is over nonnegative i.i.d.\ random variables $X_1,\ldots,X_k$. We also characterize all maximizing distributions. Using this result, we prove the Erdős Matching Conjecture for every fixed $k\ge3$, all sufficiently large $s$, and every $n\ge k(s+1)$. More specifically, we show that in this range, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $s+1$ pairwise disjoint edges is \[ \max\left\{ \binom{k(s+1)-1}{k}, \binom nk-\binom{n-s}{k} \right\}. \] Moreover, every extremal family consists either of all $k$-sets meeting a fixed $s$-set or of all $k$-sets contained in a fixed $(k(s+1)-1)$-set.

We prove a regularity lemma for polynomials with fixed degree $d$ and any number of standard Gaussian input variables. Following modern approaches to the Szemerédi regularity lemma, we deduce our results from a general formulation of the regularity lemma using ultraproducts. Along the way, we give a direct combinatorial proof of the celebrated “fourth moment theorem” of Nualart and Peccati, which characterizes the asymptotically Gaussian polynomials inside degree-$d$ Gaussian chaos as those such that $\mathbb{E} p^2 \to 1$ and $\mathbb{E} p^4 \to 3$.

Let $\mu_q$ be the distribution of the normalized Kloosterman sums $\mathrm{Kl}(a;q)$, $a\in\mathbb{F}_q^\times$, and let $\mu_{\mathrm{ST}}$ be the Sato–Tate law. We prove $W_s(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/3}$ for $1\le s\le2$, $W_s(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/6-1/(3s)}$ for $2\le s<\infty$, and $W_\infty(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/6}$, answering a question of Kowalski and Untrau. More generally, we bound $W_s(\mu,\mu_{\mathrm{ST}})$ for all $s$ whenever the Chebyshev moments of $\mu$ satisfy bounds of Katz type. For $s=2$ the bound also follows from a smoothing inequality of Borda and Cuenin; for $s>2$ and $s=\infty$ our bounds are stronger. The proof uses the quantile coupling, which arises from Stieltjes sums over two different intervals whose cells are paired so as to balance two measures, and which is optimal for all $s$ simultaneously; an explicit $W_2$ smoothing inequality relative to $\mu_{\mathrm{ST}}$; and a sharp comparison between $W_s$ and the Kolmogorov distance for laws with square-root edges. The last result shows that $q^{-1/6}$ is the best rate for $W_\infty$ obtainable from Niederreiter's discrepancy bound. Numerically, for primes $10^3<p<10^6$, all these distances appear to decay like $p^{-1/2}$.

We prove the Fourier entropy–influence inequality for every linear threshold function, with a constant below $8.937$ and no restrictions on its weights or threshold. The argument applies to regular Boolean functions and gives logarithmic bounds for their Boolean combinations and for unions with bounded variable overlap. Character multiplication preserves the estimates and yields bounds by influence minus variance. Under arbitrary product measures, symmetric thresholds satisfy the strengthened inequality needed for read-once composition, uniformly in depth and fan-in. A joint description of each Fourier set refines the overlap estimate and recovers constant bounds for addressing. Exact selector identities extend the bound to orthogonal multiplexers with shared data. Rare activation preserves the optimal constant of unrestricted FEI. The proofs combine pivotal antichains, spectral reference measures, and product-measure composition.

The elephant random walk (ERW) is a discrete-time random walk with complete memory. At every step the walker chooses one of the previous increments at random and then either repeats with a constant probability $p\in [0,1]$ or reverses it. Since its introduction by Schütz and Trimper, the model has become a useful probabilistic framework for studying long-range dependence, anomalous diffusion, reinforcement, and phase transitions. This paper proposes a generalized ERW model with time-dependent memory. In this generalized model, the usual constant memory parameter $p$ in the standard ERW is allowed to change with time. The cental limit theorems, and the law of the iterated logarithm and some other strong laws are obtained.

The median-of-means (MoM) estimator is a robust alternative to the sample mean for heavy-tailed or contaminated data. While its non-asymptotic properties are well understood for i.i.d.\ samples, much less is known for dependent sequences. We establish non-asymptotic deviation bounds for the MoM estimator computed from consecutive blocks of an identically distributed $\alpha$-mixing sequence, both for a single mean and uniformly over a class of functions (the median-of-means process). All bounds remain valid, with explicitly adjusted constants, when a sufficiently small fraction of the blocks is adversarially corrupted. As applications, we obtain deviation bounds for a median-of-means Nadaraya–Watson regression estimator, for the MoM kernel density estimator, and for empirical and Bahadur-type estimators of Value-at-Risk and Conditional Value-at-Risk. Finally, we show that the sampling of bulk materials, where the measurements form a spatially dependent random field, reduces to the one-dimensional setting if the samples are taken in well-separated groups.

In this short note we give a Khintchine-type inequality involving non-independentrandom variables. The basic setting is inspired by Breuer-Major theorem and the proof is based on a hypercontractivity argument.

We study particle approximations of maximum mean discrepancy (MMD) flows with the Coulomb kernel on the torus $\mathbb{T}^d$, $d\ge2$, driven by thermal noise of strength $\tau\ge0$. For smooth, strictly positive initial and target densities and i.i.d. initial positions, we prove that the expected Wasserstein-1 distance between the empirical measure and the mean-field solution is at most of order $N^{-1/d}$ for $d\ge3$ and $\sqrt{(\log N)/N}$ for $d=2$, uniformly in time and in the temperature $\tau\ge0$, including the deterministic case $\tau=0$. At each fixed $\tau>0$, we also prove propagation of chaos in total variation for any $k$-fold marginal distributions of particles, uniformly in time, with a bound of order $\sqrt{k/\tau}$ times the empirical convergence rate. Our proof uses the modulated free-energy method of Bresch, Jabin, and Wang to reduce the particle convergence estimate to a relaxation estimate that controls the time integral of the mean-field velocity gradient uniformly in $\tau\ge0$. Near zero temperature, where the smoothing effect of the noise degenerates, we obtain this estimate by recentering the dynamics at thermal equilibrium and using the damping generated by the Coulomb interaction.

In Fermi's 1926 note, the expression assigned to the probability that all partial sums remain below a threshold is the complementary crossing probability. We obtain the correct probability from the same diffusion equation and locate the inconsistency in the final integration. We then state the limit for independent increments under the Lindeberg condition. A counterexample with smooth densities shows that small individual variances do not suffice to justify this limit. The probabilistic results are classical; the purpose of this note is to reconcile the stated event with the printed calculation and to make the sufficient hypotheses for the discrete limit explicit.

Based on a semigroup approach, we prove quantitative uniform-in-time propagation of chaos for two systems of interacting particles with singular interaction kernels $K$ of type $L^k$, $k>d$, and a contractive Lipschitz confinement force field. First, we consider a system with moderate interaction, which means that the particles interact through the mollified kernel $K\ast V^N$, for some approximation of identity $V^N = N^{d\alpha} V(N^\alpha \cdot),\, \alpha>0$. Assuming that $\|K\|_{L^k}$ is sufficiently small, we obtain that for any $\varepsilon>0$ the uniform-in-time rate is of order $N^{-\min\left(\alpha,\frac12-\frac{d\alpha}{2}-\varepsilon\right)}$, plus an exponentially small in time contribution of the initial condition. Second, for the classical, non-mollified particle system we establish uniform-in-time propagation of chaos with rate $N^{-\min\left(\alpha,\frac12-\frac{d\alpha}{2}-\varepsilon\right)} + \|K-K\ast V^N\|_{L^{k}}$, plus an exponentially small in time contribution of the initial condition. The method relies on the regularity properties of the solution of the associated nonlinear Fokker-Planck equation combined with estimates of the Duhamel formulation of the mollified empirical measure, together with partial Girsanov transforms for the non-mollified system of particles.

We establish sharp weighted Rogers–Shephard inequalities for Rényi differential entropy. For independent copies $X,X'$ of a log-concave random vector in $\R^n$, we prove $h_p(aX-bX')\le h_p(X)+n\log(a+b)$ for all $a,b>0$ and $1\le p\le\infty$, where orders $1$ and $\infty$ correspond to Shannon entropy and min-entropy, respectively. The Shannon case settles the entropic Rogers–Shephard conjecture of Madiman and Kontoyiannis and its higher-dimensional formulation. For $1\le p<\infty$, equality holds precisely for invertible affine images of random vectors with independent exponential coordinates. At $p=\infty$, the larger equality class is characterized by possibly extended-valued convex potentials that are positively homogeneous of degree one. For every $0<p<1$, uniform laws on sufficiently high-dimensional simplices violate the unweighted bound. Thus $p=1$ is the critical order for this bound to hold in every dimension. The proof combines weighted convex intersections with an exponential lift to a log-affine probability density. At the Shannon endpoint, this lift yields an exact deficit decomposition with a conditional mutual information term for an additive channel with asymmetric Laplace noise.

We establish uniform score bounds and Lipschitz continuity for truncated Gaussian, Cauchy, Boltzmann, and softmax policies under explicitly stated parameterizations.

A random geometric graph is a random graph model whose vertex set is a binomial point process, where two vertices are connected by an edge if each lies within the ball of radius $r$ centered at the other. In this paper, we investigate the asymptotic fluctuations of the threshold radius required for the maximum degree of the graph to reach a target value. Assuming this target value grows with the number of vertices $n$ at an order of at least $\log n$ but strictly less than $(\log n)^d$, we show that these fluctuations follow a Gumbel distribution. We also investigate the spatial configurations of vertices with extremely large degrees, and show that the point process marked by their degrees converges in distribution to a Poisson point process.

FirstFit colors each arriving vertex with the smallest positive integer absent from its previously colored neighborhood. We prove that this rule can require $n^{\Omega(1)}$ many colors on certain bipartite graph of $n$ vertices even when the vertices arrive in a uniformly random order. This refutes the conjecture of Bosek, Gutowski, Lasoń, and Przybyło [BGLP24] that FirstFit uses a polylogarithmic expected number of colors on every bipartite graph in the random order model. We obtain the lower bound by analyzing the FirstFit coloring process on random bipartite graphs, where vertices arrive to either side with equal probability, and with an independent edge density $n^{-x}$ for some $0 < x < 1$. Our result can also be viewed as a qualitative strengthening of Kučera [Kuč91], who showed that, for every $\epsilon > 0$, there are graphs with chromatic number at most $n^{\epsilon}$ on which FirstFit uses $\Omega(n/\log n)$ colors in expectation under a uniformly random order. Our result shows that such poor behavior of FirstFit already occurs for $\chi(G) = 2$.

We derive a quantitative central limit theorem under sublinear expectations from a spatial endpoint regularity estimate for the one-dimensional \(G\)-heat equation. For bounded Lipschitz initial data, the second spatial derivative and the time derivative of its solution are Lipschitz in space and \(1/2\)-Hölder in time away from the initial time. In particular, \([u_{xx}(t,\cdot)]_{\mathrm{Lip}}\le C_G\Lip(\phi)/t\). Combining this estimate with the nonlinear Stein identity introduced in Song \cite{Song2020} and a replacement argument yields a bound of order \((1+\log n)/\sqrt n\) for bounded Lipschitz tests when the third absolute moment is finite and the lower variance is positive. The final replacement step is estimated directly, avoiding the nonintegrable singularity at time zero.

Gaussian mixtures can have more modes than components. For three Gaussian components with a common covariance, four modes are attainable, whereas the best general upper bound was eight. We prove that four is the exact maximum in every dimension $d \ge 2$, for arbitrary means and positive weights. We further show that the sharp bound drops to three when the whitened means are collinear or form a right or obtuse triangle. The result covers degenerate modes without genericity or separate finiteness assumptions. Our proof maps modes to local minima of a separable free-energy functional on the probability simplex, reduces the problem to counting scalar inverse-function branches, and controls competing branches through curvature and stationary-height comparisons.

We study invariant probabilities for nonlinear two-population branching systems with competition, cooperation, or predator–prey interaction, driven by Brownian noise and spectrally positive stable jumps. Without immigration, every invariant probability of a conservative competitive or predator–prey system is concentrated on the boundary under either boundary convention. With nontrivial axis dynamics, coordinatewise continuation admits only the point mass at the origin. For cooperative systems, we give criteria for boundary concentration and interior stationarity, including a sharp criterion for bilinear cooperation with a common relative noise exponent. With immigration, explicit parameter conditions yield existence and stationary moment bounds. For square-root diffusions with positive immigration, these conditions ensure interior concentration and full support on the closed quadrant. Further lower bounds on immigration rates give uniqueness and convergence of time-averaged transition laws.

Let $Z_n^i=(Z_n^i(1),\cdots,Z_n^i(d)), n\geqslant0$, be a supercritical $d$-type branching process starting from a single particle of type $i\in\{1,\cdots,d\}$, where $d$ is a positive integer. The Kesten-Stigum theorem states that if the mean matrix $M$ of the offspring distribution has spectral radius $\rho>1$ with the associated positive left and right eigenvectors $u$ and $v$, then for all $1\leqslant i \leqslant d$, the sequence $W_n^i=\langle Z_n^i,u\rangle/\rho^n u(i)$ forms a non-negative martingale, and hence converges almost surely to a random variable $W^i$ as $n\rightarrow\infty$. In this paper, we investigate the asymptotic properties of the martingale $\{W_n^i\}$, focusing on weighted moments and sharp rates of convergence. We establish necessary and sufficient conditions for the finiteness of $\phi$-moments of $W^i$, where $\phi(x)=x^{\alpha}\ell(x)$ with $\alpha\geqslant1$ and $\ell$ is a positive function slowly varying at infinity. We also determine the sharp exponential rate of convergence of $W_n^i$ to $W^i$ in the $\phi$-moment sense, identify a phase transition at the critical rate $\rho^{1/2},$ and characterize the behavior at the critical value. Our proofs rely on the double martingale structure of the fundamental martingale and on convex inequalities for martingales.