Probability

New submissions

• New submissions
• Cross-lists
• Replacements

New submissions for Fri, 31 May 24

[1]  arXiv:2405.19382 [pdf, ps, other]

Title: Decoupling and Multipoint moments for the Inverse of the Gaussian multiplicative chaos

Authors: Ilia Binder, Tomas Kojar

Comments: arXiv admin note: substantial text overlap with arXiv:2305.00360

Subjects: Probability (math.PR)

In this article we study the decoupling structure and multipoint moment of the inverse of the Gaussian multiplicative chaos. It is also the second part of preliminary work for extending the work in "Random conformal weldings" (by K. Astala, P. Jones, A. Kupiainen, E. Saksman) to the existence of Lehto welding for the inverse. In particular, we prove that the dilatation of the inverse homeomorphism on the positive real line is in $L^{1}([0,1]\times[0,2])$.

[2]  arXiv:2405.19437 [pdf, ps, other]

Title: Quantitative hydrodynamics for a generalized contact model

Authors: Julian Amorim, Milton Jara, Yangrui Xiang

Subjects: Probability (math.PR); Mathematical Physics (math-ph); Biological Physics (physics.bio-ph)

We derive a quantitative version of the hydrodynamic limit for an interacting particle system inspired by integrate-and-fire neuron models. More precisely, we show that the $L^2$-speed of convergence of the empirical density of states in a generalized contact process defined over a $d$-dimensional torus of size $n$ is of the optimal order $\mathcal O(n^{d/2})$. In addition, we show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by a inhomogeneous stochastic linear equation.

[3]  arXiv:2405.19482 [pdf, ps, other]

Title: Malliavian differentiablity and smoothness of density for SDES with locally Lipschitz coefficients

Authors: Cristina Anton

Comments: submitted to Stochastic Analysis and Applications

Subjects: Probability (math.PR)

We study Malliavin differentiability for the solutions of a stochastic differential equation with drift of super-linear growth. Assuming we have a monotone drift with polynomial growth, we prove Malliavin differentiability of any order. As a consequence of this result, under the H\"ormander's hypothesis we prove that the density of the solution's law with respect to the Lebesgue measure is infinitely differentiable. To avoid non-integrability problems due to the unbounded drift, we follow an approach based on the concepts of Ray Absolute Continuity and Stochastic Gate\^aux Differentiability.

[4]  arXiv:2405.19541 [pdf, ps, other]

Title: The pivotal set of a Boolean function

Authors: Raphaël Cerf

Subjects: Probability (math.PR)

We define the pivotal set of a Boolean function and we prove a fundamental inequality on its expected size, when the inputs are independent random coins of parameter~$p$. We give two complete proofs of this inequality. Along the way, we obtain the classical Margulis--Russo formula. We give a short proof of the classical Hoeffding inequality for i.i.d. Bernoulli random variables, and we use it to derive more complex deviations inequalities associated to the pivotal set. We follow finally Talagrand's footsteps and we discuss a beautiful inequality that he proved in the uniform case.

[5]  arXiv:2405.19756 [pdf, other]

Title: Upper deviation probabilities for the range of a supercritical super-Brownian motion

Authors: Shuxiong Zhang

Comments: 8

Subjects: Probability (math.PR)

Let $\{X_t\}_{t\geq 0 }$ be a $d$-dimensional supercritical super-Brownian motion started from the origin with branching mechanism $\psi$. Denote by $R_t:=\inf\{r>0:X_s(\{x\in \mathbb{R}^d:|x|\geq r\})=0,~\forall~0\leq s\leq t\}$ the radius of the minimal ball (centered at the origin) containing the range of $\{X_s\}_{s\geq 0 }$ up to time $t$. In \cite{Pinsky}, Pinsky proved that condition on non-extinction, $\lim_{t\to\infty}R_t/t=\sqrt{2\beta}$ in probability, where $\beta:=-\psi'(0)$. Afterwards, Engl\"{a}nder \cite{Englander04} studied the lower deviation probabilities of $R_t$. For the upper deviation probabilities, he \cite[Conjecture 8]{Englander04} conjectured that for $\rho>\sqrt {2\beta}$,
$$ \lim_{t\to\infty}\frac{1}{t}\log\mathbb{P}(R_t\geq \rho t)=-\left(\frac{\rho^2}{2}-\beta\right). $$ In this note, we confirmed this conjecture.

[6]  arXiv:2405.19780 [pdf, ps, other]

Title: Convergence Results for Approximation with independent Variables

Authors: Freddy Delbaen, Chitro Majumdar

Subjects: Probability (math.PR)

For a square integrable $m$-dimensional random variable $X$ on a probability space $(\Omega,\Fc,\Pr)$ and a sub sigma algebra $\Ac$, we show that there is a constructive way to represent $X-\Er[X\mid\Ac]$ as the sum of a series of variables that are independent of $\Ac$.

[7]  arXiv:2405.19828 [pdf, other]

Title: From classical to modern central limit theorems

Authors: Vladimir V. Ulyanov

Comments: 23 pages, 6 figures

Subjects: Probability (math.PR)

De Moivre (1733), investigating the limit distribution of the binomial distribution, was the first to discover the existence of the normal distribution and the central limit theorem. In this review article, we briefly recall the history of classical central limit theorem and martingale central limit theorem, and introduce a new direction of central limit theorem, namely nonlinear central limit theorem and nonlinear normal distribution.

[8]  arXiv:2405.19881 [pdf, ps, other]

Title: (Non)-hyperuniformity of perturbed lattices

Authors: David Dereudre, Daniela Flimmel, Martin Huesmann, Thomas Leblé

Comments: 53p. Comments welcome

Subjects: Probability (math.PR)

We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite $d$-moment, and that this condition is sharp. When $d \geq 3$, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform. As a side remark of independent interest, we exhibit hyperuniform processes with arbitrarily slow decay of their number variance.

[9]  arXiv:2405.19903 [pdf, other]

Title: On a new family of weighted Gaussian processes: an application to bat telemetry data

Authors: Jose Hermenegildo Ramirez Gonzalez, Antonio Murillo Salas, Ying Sun

Comments: 26 pages, 7 Figures, 2 Tables

Subjects: Probability (math.PR); Statistics Theory (math.ST)

In this article we use a covariance function that arises from limit of fluctuations of the rescaled occupation time process of a branching particle system, to introduce a family of weighted long-range dependence Gaussian processes. In particular, we consider two subfamilies for which we show that the process is not a semimartingale, that the processes exhibit long-range dependence and have long-range memory of logarithmic order. Finally, we illustrate that this family of processes is useful for modeling real world data.

[10]  arXiv:2405.19926 [pdf, ps, other]

Title: Invariant Measure for Linear Stochastic PDEs in the space of Tempered distributions

Authors: Arvind Kumar Nath

Subjects: Probability (math.PR)

In this paper, we first explore exponential stability by using Monotonicity inequality and use this information to obtain the existence of Invariant measure for linear Stochastic PDEs with potential in the space of tempered distributions. The uniqueness of Invariant Measure follows from Monotonicity inequality.

[11]  arXiv:2405.20019 [pdf, ps, other]

Title: Zeros of the Brownian Sheet

Authors: Keming Chen, Guillaume Woessner

Comments: 22 pages, 4 sections

Subjects: Probability (math.PR)

In this work we firstly answer to a question raised by Khoshnevisan in \cite[Open Problem 4]{khoshnevisan2007slices} by proving that almost surely there is no projection of big enough rank changing the Hausdorff dimension of the zeros of the Brownian sheet. Secondly, we prove that almost surely for every projection whose rank isn't matching the aforementioned condition, the projection of the zero set is the entirety of the projective space.\\ \textbf{Key words:} Brownian sheet, zeros set, Hausdorff dimension, orthogonal projection.

[12]  arXiv:2405.20087 [pdf, ps, other]

Title: Characterization of probability distributions on some locally compact Abelian groups containing an element of order 2

Authors: Gennadiy Feldman

Comments: 16 pages

Subjects: Probability (math.PR)

The well-known Heyde theorem characterizes the Gaussian distributions on the real line by the symmetry of the conditional distribution of one linear form of independent random variables given another. We generalize this theorem to groups of the form $\mathbb{R}\times F$, where $F$ is a finite Abelian group such that its 2-component is isomorphic to the additive group of the integers modulo $2$. In so doing, coefficients of the linear forms are arbitrary topological automorphisms of the group. Previously, a similar result was proved in the case when the group $F$ contains no elements of order 2. The presence of an element of order 2 in $F$ leads to the fact that a new class of probability distributions is characterized

[13]  arXiv:2405.20148 [pdf, ps, other]

Title: SLE and its partition function in multiply connected domains via the Gaussian Free Field and restriction measures

Authors: Juhan Aru, Philémon Bordereau

Comments: 30 pages, 4 figures

Subjects: Probability (math.PR); Mathematical Physics (math-ph)

One way to uniquely define Schramm-Loewner Evolution (SLE) in multiply connected domains is to use the restriction property. This gives an implicit definition of a $\sigma$-finite measure on curves; yet it is in general not clear how to construct such measures nor whether the mass of these measures, called the partition function, is finite.
We provide an explicit construction of the such conformal restriction SLEs in multiply connected domains when $\kappa = 4$ using the Gaussian Free Field (GFF). In particular, both when the target points of the curve are on the same or on distinct boundary components, we show that there is a mixture of laws of level lines of GFFs that satisfies the restriction property. This allows us to give an expression for the partition function of $\mathrm{sle}_4$ on multiply connected domains and shows that the partition function is finite, answering the question raised in [Lawler, J. Stat. Phys. 2009].
In a second part, we provide a second construction of $\mathrm{sle}_\kappa$ in multiply-connected domains for the whole range $\kappa \in (8/3,4]$; specific, however, to the case of the two target points belonging to the same boundary components. This is inspired by [Werner, Wu, Electron. J. Probab. 2013] and consists of a mixture of laws on curves obtained by following $\mathrm{cle}_\kappa$ loops and restriction hulls attached to parts of the boundary of the domain. In this case as well, we obtain as a corollary the finiteness of the partition function for this type of $\mathrm{sle}_\kappa$.

[14]  arXiv:2405.20276 [pdf, ps, other]

Title: Proof of the Diaconis--Freedman Conjecture on partially-exchangeable processes

Authors: Noah Halberstam, Tom Hutchcroft

Comments: 20 pages

Subjects: Probability (math.PR)

We prove a conjecture of Diaconis and Freedman (Ann. Probab. 1980) characterising the extreme points of the set of partially-exchangeable processes on a countable set. More concretely, we prove that the partially exchangeable sigma-algebra of any transient partially exchangeable process $X=(X_i)_{i\geq 0}$ (and hence any transient Markov chain) coincides up to null sets with the sigma-algebra generated by the initial state $X_0$ and the transition counts $( \#\{i\geq 0: X_i=x, X_{i+1}=y\} : x,y\in S)$. Our proof is based on an analysis of Gibbs measures for Eulerian paths on rooted digraphs, relying in particular on the connection to uniform spanning trees and Wilson's algorithm via the de Bruijn--Ehrenfest--Smith--Tutte (BEST) bijection, and yields an explicit method to sample from the conditional distribution of a transient Markov chain given its transition counts.

[15]  arXiv:2405.20284 [pdf, other]

Title: Fock's dimer model on the Aztec diamond

Authors: Cédric Boutillier, Béatrice de Tilière

Comments: 51 pages, 18 figures. Preliminary version

Subjects: Probability (math.PR); Combinatorics (math.CO)

We consider the dimer model on the Aztec diamond with Fock's weights, which is gauge equivalent to the model with any choice of positive weight function. We prove an explicit, compact formula for the inverse Kasteleyn matrix, thus extending numerous results in the case of periodic graphs. We also show an explicit product formula for the partition function; as a specific instance of the genus 0 case, we recover Stanley's formula. We then use our explicit formula for the inverse Kasteleyn matrix to recover, in a simple way, limit shape results; we also obtain new ones. In doing so, we extend the correspondence between the limit shape and the amoeba of the corresponding spectral curve of arXiv:2306.07482 to the case of non-generic weights.

[16]  arXiv:2405.20300 [pdf, ps, other]

Title: Kemeny's constant and the Lemoine point of a simplex

Authors: Karel Devriendt

Comments: 8 pages, 1 figure

Subjects: Probability (math.PR)

Kemeny's constant is an invariant of discrete-time Markov chains, equal to the expected number of steps between two states sampled from the stationary distribution. It appears in applications as a concise characterization of the mixing properties of a Markov chain and has many alternative definitions. In this short article, we derive a new geometric expression for Kemeny's constant, which involves the distance between two points in a simplex associated to the Markov chain: the circumcenter and the Lemoine point. Our proof uses an expression due to Wang, Dubbeldam and Van Mieghem of Kemeny's constant in terms of effective resistances and Fiedler's interpretation of effective resistances as edge lengths of a simplex.

[17]  arXiv:2405.20308 [pdf, ps, other]

Title: On the Spielman-Teng Conjecture

Authors: Ashwin Sah, Julian Sahasrabudhe, Mehtaab Sawhney

Comments: 27 pages

Subjects: Probability (math.PR); Combinatorics (math.CO)

Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $\sigma_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( \sigma_{n}(M) \leq \varepsilon n^{-1/2} \big) = (1+o(1)) \varepsilon + e^{-\Omega(n)}\] for all $0 \leq \varepsilon \ll 1$. This resolves, up to a $1+o(1)$ factor, a seminal conjecture of Spielman and Teng.

Cross-lists for Fri, 31 May 24

[18]  arXiv:2405.19493 (cross-list from cs.DS) [pdf, other]

Title: Fast Gaussian Distributed Pseudorandom Number Generation in Java via the Ziggurat Algorithm

Authors: Vincent A. Cicirello

Subjects: Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Probability (math.PR)

We report on experiments with the ziggurat algorithm for generating Gaussian distributed random numbers. The study utilizes our open source Java implementation that was introduced originally for Java 11 at a time when the Java API only provided the much slower polar method. Our Java implementation of the ziggurat algorithm is a port of the GNU Scientific Library's C implementation. Java 17 introduced a significant overhaul of pseudorandom number generation, including several modern pseudorandom number generators (PRNGs) as well as additional functionality, among which includes switching from the polar method to a modified ziggurat algorithm. In the experiments of this paper, we explore whether there is still a need for our implementation for Java 17+ applications. Our results show that Java 17's modified ziggurat is faster than our implementation for the PRNGs that support it. However, Java 17+ continues to use the polar method for the legacy PRNGs Random, SecureRandom, and ThreadLocalRandom. The linear congruential method of Java's Random class lacks the statistical properties required by Java's modified ziggurat implementation; and SecureRandom and ThreadLocalRandom unfortunately use the polar method as a side-effect of extending Random. Our implementation of the original ziggurat algorithm does not require the same statistical properties of the underlying PRNG as Java 17's optimized version, and can be used with any of these PRNGs, and is especially relevant where pre-Java 17 support is required.

[19]  arXiv:2405.19553 (cross-list from math.ST) [pdf, ps, other]

Title: Convergence Bounds for Sequential Monte Carlo on Multimodal Distributions using Soft Decomposition

Authors: Holden Lee, Matheau Santana-Gijzen

Subjects: Statistics Theory (math.ST); Machine Learning (cs.LG); Probability (math.PR); Machine Learning (stat.ML)

We prove bounds on the variance of a function $f$ under the empirical measure of the samples obtained by the Sequential Monte Carlo (SMC) algorithm, with time complexity depending on local rather than global Markov chain mixing dynamics. SMC is a Markov Chain Monte Carlo (MCMC) method, which starts by drawing $N$ particles from a known distribution, and then, through a sequence of distributions, re-weights and re-samples the particles, at each instance applying a Markov chain for smoothing. In principle, SMC tries to alleviate problems from multi-modality. However, most theoretical guarantees for SMC are obtained by assuming global mixing time bounds, which are only efficient in the uni-modal setting. We show that bounds can be obtained in the truly multi-modal setting, with mixing times that depend only on local MCMC dynamics.

[20]  arXiv:2405.19798 (cross-list from math-ph) [pdf, ps, other]

Title: Mixed radix numeration bases: Hörner's rule, Yang-Baxter equation and Furstenberg's conjecture

Authors: Damien Simon

Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Number Theory (math.NT); Probability (math.PR); Exactly Solvable and Integrable Systems (nlin.SI)

Mixed radix bases in numeration is a very old notion but it is rarely studied on its own or in relation with concrete problems related to number theory. Starting from the natural question of the conversion of a basis to another for integers as well as polynomials, we use mixed radix bases to introduce two-dimensional arrays with suitable filling rules. These arrays provide algorithms of conversion which uses only a finite number of euclidean division to convert from one basis to another; it is interesting to note that these algorithms are generalizations of the well-known H\"orner's rule of quick evaluation of polynomials. The two-dimensional arrays with local transformations are reminiscent from statistical mechanics models: we show that changes between three numeration basis are related to the set-theoretical Yang-Baxter equation and this is, up to our knowledge, the first time that such a structure is described in number theory. As an illustration, we reinterpret well-known results around Furstenberg's conjecture in terms of Yang-Baxter transformations between mixed radix bases, hence opening the way to alternative approaches.

[21]  arXiv:2405.19807 (cross-list from cs.LG) [pdf, ps, other]

Title: MetaCURL: Non-stationary Concave Utility Reinforcement Learning

Authors: Bianca Marin Moreno (UGA, Thoth, EDF R&D, FiME Lab), Margaux Brégère (LPSM, EDF R&D), Pierre Gaillard (UGA, Thoth), Nadia Oudjane (EDF R&D, FiME Lab)

Subjects: Machine Learning (cs.LG); Probability (math.PR); Statistics Theory (math.ST); Machine Learning (stat.ML)

We explore online learning in episodic loop-free Markov decision processes on non-stationary environments (changing losses and probability transitions). Our focus is on the Concave Utility Reinforcement Learning problem (CURL), an extension of classical RL for handling convex performance criteria in state-action distributions induced by agent policies. While various machine learning problems can be written as CURL, its non-linearity invalidates traditional Bellman equations. Despite recent solutions to classical CURL, none address non-stationary MDPs. This paper introduces MetaCURL, the first CURL algorithm for non-stationary MDPs. It employs a meta-algorithm running multiple black-box algorithms instances over different intervals, aggregating outputs via a sleeping expert framework. The key hurdle is partial information due to MDP uncertainty. Under partial information on the probability transitions (uncertainty and non-stationarity coming only from external noise, independent of agent state-action pairs), we achieve optimal dynamic regret without prior knowledge of MDP changes. Unlike approaches for RL, MetaCURL handles full adversarial losses, not just stochastic ones. We believe our approach for managing non-stationarity with experts can be of interest to the RL community.

[22]  arXiv:2405.19995 (cross-list from stat.ML) [pdf, other]

Title: Symmetries in Overparametrized Neural Networks: A Mean-Field View

Authors: Javier Maass Martínez, Joaquin Fontbona

Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR)

We develop a Mean-Field (MF) view of the learning dynamics of overparametrized Artificial Neural Networks (NN) under data symmetric in law wrt the action of a general compact group $G$. We consider for this a class of generalized shallow NNs given by an ensemble of $N$ multi-layer units, jointly trained using stochastic gradient descent (SGD) and possibly symmetry-leveraging (SL) techniques, such as Data Augmentation (DA), Feature Averaging (FA) or Equivariant Architectures (EA). We introduce the notions of weakly and strongly invariant laws (WI and SI) on the parameter space of each single unit, corresponding, respectively, to $G$-invariant distributions, and to distributions supported on parameters fixed by the group action (which encode EA). This allows us to define symmetric models compatible with taking $N\to\infty$ and give an interpretation of the asymptotic dynamics of DA, FA and EA in terms of Wasserstein Gradient Flows describing their MF limits. When activations respect the group action, we show that, for symmetric data, DA, FA and freely-trained models obey the exact same MF dynamic, which stays in the space of WI laws and minimizes therein the population risk. We also give a counterexample to the general attainability of an optimum over SI laws. Despite this, quite remarkably, we show that the set of SI laws is also preserved by the MF dynamics even when freely trained. This sharply contrasts the finite-$N$ setting, in which EAs are generally not preserved by unconstrained SGD. We illustrate the validity of our findings as $N$ gets larger in a teacher-student experimental setting, training a student NN to learn from a WI, SI or arbitrary teacher model through various SL schemes. We last deduce a data-driven heuristic to discover the largest subspace of parameters supporting SI distributions for a problem, that could be used for designing EA with minimal generalization error.

[23]  arXiv:2405.20086 (cross-list from math.ST) [pdf, other]

Title: Analysis of a multi-target linear shrinkage covariance estimator

Authors: Benoit Oriol

Subjects: Statistics Theory (math.ST); Machine Learning (cs.LG); Probability (math.PR); Machine Learning (stat.ML)

Multi-target linear shrinkage is an extension of the standard single-target linear shrinkage for covariance estimation. We combine several constant matrices - the targets - with the sample covariance matrix. We derive the oracle and a \textit{bona fide} multi-target linear shrinkage estimator with exact and empirical mean. In both settings, we proved its convergence towards the oracle under Kolmogorov asymptotics. Finally, we show empirically that it outperforms other standard estimators in various situations.

[24]  arXiv:2405.20116 (cross-list from math.OC) [pdf, ps, other]

Title: Complexity of Zeroth- and First-order Stochastic Trust-Region Algorithms

Authors: Yunsoo Ha, Sara Shashaani, Raghu Pasupathy

Subjects: Optimization and Control (math.OC); Numerical Analysis (math.NA); Probability (math.PR)

Model update (MU) and candidate evaluation (CE) are classical steps incorporated inside many stochastic trust-region (TR) algorithms. The sampling effort exerted within these steps, often decided with the aim of controlling model error, largely determines a stochastic TR algorithm's sample complexity. Given that MU and CE are amenable to variance reduction, we investigate the effect of incorporating common random numbers (CRN) within MU and CE on complexity. Using ASTRO and ASTRO-DF as prototype first-order and zeroth-order families of algorithms, we demonstrate that CRN's effectiveness leads to a range of complexities depending on sample-path regularity and the oracle order. For instance, we find that in first-order oracle settings with smooth sample paths, CRN's effect is pronounced -- ASTRO with CRN achieves $\tilde{O}(\epsilon^{-2})$ a.s. sample complexity compared to $\tilde{O}(\epsilon^{-6})$ a.s. in the generic no-CRN setting. By contrast, CRN's effect is muted when the sample paths are not Lipschitz, with the sample complexity improving from $\tilde{O}(\epsilon^{-6})$ a.s. to $\tilde{O}(\epsilon^{-5})$ and $\tilde{O}(\epsilon^{-4})$ a.s. in the zeroth- and first-order settings, respectively. Since our results imply that improvements in complexity are largely inherited from generic aspects of variance reduction, e.g., finite-differencing for zeroth-order settings and sample-path smoothness for first-order settings within MU, we anticipate similar trends in other contexts.

[25]  arXiv:2405.20250 (cross-list from math.OC) [pdf, ps, other]

Title: Entropy annealing for policy mirror descent in continuous time and space

Authors: Deven Sethi, David Šiška, Yufei Zhang

Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Probability (math.PR)

Entropy regularization has been extensively used in policy optimization algorithms to regularize the optimization landscape and accelerate convergence; however, it comes at the cost of introducing an additional regularization bias. This work quantifies the impact of entropy regularization on the convergence of policy gradient methods for stochastic exit time control problems. We analyze a continuous-time policy mirror descent dynamics, which updates the policy based on the gradient of an entropy-regularized value function and adjusts the strength of entropy regularization as the algorithm progresses. We prove that with a fixed entropy level, the dynamics converges exponentially to the optimal solution of the regularized problem. We further show that when the entropy level decays at suitable polynomial rates, the annealed flow converges to the solution of the unregularized problem at a rate of $\mathcal O(1/S)$ for discrete action spaces and, under suitable conditions, at a rate of $\mathcal O(1/\sqrt{S})$ for general action spaces, with $S$ being the gradient flow time. This paper explains how entropy regularization improves policy optimization, even with the true gradient, from the perspective of convergence rate.

[26]  arXiv:2405.20311 (cross-list from math.NT) [pdf, other]

Title: Martingale central limit theorem for random multiplicative functions

Authors: Ofir Gorodetsky, Mo Dick Wong

Comments: 41 pages, 2 figures; comments are welcome

Subjects: Number Theory (math.NT); Probability (math.PR)

Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we prove that the sum $\sum_{n \le x}\alpha(n) f(n)$, normalised to have mean square $1$, has a limiting distribution. The limiting distribution we find is a Gaussian times the square-root of the total mass of a random measure associated with $f$.
Our result applies to $d_z$, the $z$-th divisor function, as long as $z$ is strictly between $0$ and $\tfrac{1}{\sqrt{2}}$. Other examples of admissible $f$-s include any multiplicative indicator function with the property that $f(p)=1$ holds for a set of primes of density strictly between $0$ and $\tfrac{1}{2}$.

Replacements for Fri, 31 May 24

[27]  arXiv:2112.10739 (replaced) [pdf, other]

Title: A diagram-free approach to the stochastic estimates in regularity structures

Authors: Pablo Linares, Felix Otto, Markus Tempelmayr, Pavlos Tsatsoulis

Comments: 76 pages, 5 figures. Changes in the 3rd version: more details on the logical order of the induction, updated bibliographical context

Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

[28]  arXiv:2205.07596 (replaced) [pdf, other]

Title: Exact Exponents for Concentration and Isoperimetry in Product Polish Spaces

Authors: Lei Yu

Comments: IEEE Transactions on Information Theory

Subjects: Probability (math.PR); Information Theory (cs.IT); Functional Analysis (math.FA); Metric Geometry (math.MG)

[29]  arXiv:2212.07474 (replaced) [pdf, ps, other]

Title: A Characterization of the n-th Degree Bounded Stochastic Dominance

Authors: Bar Light, Andres Perlroth

Subjects: Probability (math.PR)

[30]  arXiv:2302.14823 (replaced) [pdf, other]

Title: Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices

Authors: Nicholas A. Cook, Raphael Ducatez, Alice Guionnet

Comments: Substantially revised. Relaxed assumptions in the main results. Added new results on the conditional eigenvector structure, Erd\H{o}s--R\'enyi graphs, and non-universality of the rate function away from the bulk. Added some numerical illustrations and a discussion of open problems. 101 pages, 2 figures

Subjects: Probability (math.PR)

[31]  arXiv:2303.18192 (replaced) [pdf, other]

Title: Characterizing models in regularity structures: a quasilinear case

Authors: Markus Tempelmayr

Comments: Minor typographical changes

Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

[32]  arXiv:2307.05407 (replaced) [pdf, other]

Title: Weyl's law in Liouville quantum gravity

Authors: Nathanaël Berestycki, Mo Dick Wong

Comments: Typos fixed; re-organised introduction with expanded literature review

Subjects: Probability (math.PR); Mathematical Physics (math-ph); Differential Geometry (math.DG); Spectral Theory (math.SP); Chaotic Dynamics (nlin.CD)

[33]  arXiv:2309.16591 (replaced) [pdf, ps, other]

Title: Preferential attachment with choice based edge-step

Authors: Yury Malyshkin

Subjects: Probability (math.PR)

[34]  arXiv:2403.18532 (replaced) [pdf, ps, other]

Title: Scaling limits for random walks on long range percolation clusters

Authors: Noam Berger, Yuki Tokushige

Comments: 32 pages, we removed a flow chart from the previous version

Subjects: Probability (math.PR)

[35]  arXiv:2404.11167 (replaced) [pdf, ps, other]

Title: Ito's formula for flows of conditional measures on semimartingales

Authors: Xin Guo, Jiacheng Zhang

Subjects: Probability (math.PR)

[36]  arXiv:2405.01045 (replaced) [pdf, ps, other]

Title: Well-posedness of stochastic mSQG equations with Kraichnan noise and $L^p$ data

Authors: Shuaijie Jiao, Dejun Luo

Comments: 33 pages. We have updated the relation of $\beta_N$ and $\beta_L$ in Lemma 2.2, following Proposition 2.7 in arXiv:2308.03216v2. Moreover, we have simplified the statements of Theorem 1.4, covering slightly wider range of parameters

Subjects: Probability (math.PR)

[37]  arXiv:2405.16764 (replaced) [pdf, ps, other]

Title: One-dimensional SRBM with discontinuous state-dependent drift and variance and its stationary distribution

Authors: Masakiyo Miyazawa

Subjects: Probability (math.PR)

[38]  arXiv:2405.18796 (replaced) [pdf, ps, other]

Title: Spectral measure of large random Helson matrices

Authors: Yanqi Qiu, Guocheng Zhen

Comments: 18 pages. arXiv admin note: text overlap with arXiv:math/0307330 by other authors

Subjects: Probability (math.PR); Number Theory (math.NT)

[39]  arXiv:1904.01048 (replaced) [pdf, other]

Title: Non-compact quantum spin chains as integrable stochastic particle processes

Authors: Rouven Frassek, Cristian Giardinà, Jorge Kurchan

Comments: 35 pages, 2 figures, v2: typos fixed and references added, v3: typo fixed, v4,5: minor correction

Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Probability (math.PR)

[40]  arXiv:2208.02874 (replaced) [pdf, other]

Title: Anticoncentration in Ramsey graphs and a proof of the Erdős-McKay conjecture

Authors: Matthew Kwan, Ashwin Sah, Lisa Sauermann, Mehtaab Sawhney

Subjects: Combinatorics (math.CO); Probability (math.PR)

[41]  arXiv:2311.04039 (replaced) [pdf, other]

Title: Free Integral Calculus

Authors: Franz Lehner, Kamil Szpojankowski

Comments: 56 pages, 4 figures; changes in v3: new example - a random walk of length 3 on the free group; some minor corrections

Subjects: Operator Algebras (math.OA); Probability (math.PR)

[42]  arXiv:2312.16214 (replaced) [pdf, ps, other]

Title: Stochastic Equilibrium the Lucas Critique and Keynesian Economics

Authors: David Staines

Comments: (More Minor Edits) Journal Submission 139 pages (Main Text) + 85 (Supplementary Materials)

Subjects: Theoretical Economics (econ.TH); Econometrics (econ.EM); Algebraic Topology (math.AT); General Topology (math.GN); Probability (math.PR)

[43]  arXiv:2401.09834 (replaced) [pdf, ps, other]

Title: Convergence of a spatial semidiscretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise

Authors: Binjie Li, Qin Zhou

Subjects: Numerical Analysis (math.NA); Probability (math.PR)

[44]  arXiv:2403.16878 (replaced) [pdf, ps, other]

Title: Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions

Authors: Bjoern Bringmann, Sky Cao

Comments: 103 pages, fixed several minor errors

Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Probability (math.PR)

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