Showing papers on "Central limit theorem published in 2020"

Bit Error Probability for Large Intelligent Surfaces Under Double-Nakagami Fading Channels

Abstract: In this work, we investigate the probability distribution function of the channel fading between a base station, an array of intelligent reflecting elements, known as large intelligent surfaces (LIS), and a single-antenna user. We assume that both fading channels, i.e., the channel between the base station and the LIS, and the channel between the LIS and the single user are Nakagami- $m$ distributed. Additionally, we derive the exact bit error probability considering quadrature amplitude ( $M$ -QAM) and binary phase-shift keying (BPSK) modulations when the number of LIS elements, $n$ , is equal to 2 and 3. We assume that the LIS can perform phase adjustment, but there is a residual phase error modeled by a Von Mises distribution. Based on the central limit theorem, and considering a large number of reflecting elements, we also present an accurate approximation and upper bounds for the bit error rate. Through several Monte Carlo simulations, we demonstrate that all derived expressions perfectly match the simulated results.

Universality Laws for High-Dimensional Learning with Random Features.

Abstract: We prove a universality theorem for learning with random features. Our result shows that, in terms of training and generalization errors, a random feature model with a nonlinear activation function is asymptotically equivalent to a surrogate linear Gaussian model with a matching covariance matrix. This settles a so-called Gaussian equivalence conjecture based on which several recent papers develop their results. Our method for proving the universality theorem builds on the classical Lindeberg approach. Major ingredients of the proof include a leave-one-out analysis for the optimization problem associated with the training process and a central limit theorem, obtained via Stein's method, for weakly correlated random variables.

The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line

Abstract: We prove that if $\omega $ is uniformly distributed on $[0,1]$, then as $T\to \infty $, $t\mapsto \zeta (i\omega T+it+1/2)$ converges to a nontrivial random generalized function, which in turn is identified as a product of a very well-behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between probabilistic number theory and the theory of multiplicative chaos—the latter is known to be connected to various branches of modern probability theory and mathematical physics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $\delta _{T}$ approach zero slowly enough as $T\to \infty $, then $t\mapsto \zeta (1/2+i\delta _{T}t+i\omega T)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg’s central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.

Monte Carlo with determinantal point processes

Abstract: We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical N^{−1/2} , where N is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures involving determinantal point processes associated with multivariate orthogonal polynomials, and we obtain root mean square errors that decrease as N^{−(1+1/d)/2} , where d is the dimension of the ambient space. First, we prove a central limit theorem (CLT) for the linear statistics of a class of determinantal point processes, when the reference measure is a product measure supported on a hypercube, which satisfies the Nevai-class regularity condition; a result which may be of independent interest. Next, we introduce a Monte Carlo method based on these determinantal point processes, and prove a CLT with explicit limiting variance for the quadrature error, when the reference measure satisfies a stronger regularity condition. As a corollary, by taking a specific reference measure and using a construction similar to importance sampling, we obtain a general Monte Carlo method, which applies to any measure with continuously derivable density. Loosely speaking, our method can be interpreted as a stochastic counterpart to Gaussian quadrature, which, at the price of some convergence rate, is easily generalizable to any dimension and has a more explicit error term.

Packets of Diffusing Particles Exhibit Universal Exponential Tails.

Abstract: Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function $P(X,t)$ of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells, and bacteria suspensions. We show that such exponential behavior is generally valid in a large class of problems of transport in random media. By extending the large deviations approach for a continuous time random walk, we uncover a general universal behavior for the decay of the density. It is found that fluctuations in the number of steps of the random walker, performed at finite time, lead to exponential decay (with logarithmic corrections) of $P(X,t)$. This universal behavior also holds for short times, a fact that makes experimental observations readily achievable.

Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices

Abstract: We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy–Widom law for the largest nonspiked sample eigenvalue is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a central limit theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.

Gaussian fluctuations for the stochastic heat equation with colored noise

Abstract: In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit theorem is described in the total variation distance, using Malliavin calculus and Stein’s method. We also provide a functional central limit theorem.

On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration

Abstract: We undertake a precise study of the asymptotic and non-asymptotic properties of stochastic approximation procedures with Polyak-Ruppert averaging for solving a linear system $\bar{A} \theta = \bar{b}$. When the matrix $\bar{A}$ is Hurwitz, we prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity. The CLT characterizes the exact asymptotic covariance matrix, which is the sum of the classical Polyak-Ruppert covariance and a correction term that scales with the step size. Under assumptions on the tail of the noise distribution, we prove a non-asymptotic concentration inequality whose main term matches the covariance in CLT in any direction, up to universal constants. When the matrix $\bar{A}$ is not Hurwitz but only has non-negative real parts in its eigenvalues, we prove that the averaged LSA procedure actually achieves an $O(1/T)$ rate in mean-squared error. Our results provide a more refined understanding of linear stochastic approximation in both the asymptotic and non-asymptotic settings. We also show various applications of the main results, including the study of momentum-based stochastic gradient methods as well as temporal difference algorithms in reinforcement learning.

Limit laws for rational continued fractions and value distribution of quantum modular forms

Abstract: We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in~$(0,1]$ ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallee on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We obtain the Gaussian behaviour of central values of the Esterman function~$\sum_{n\geq 1} \tau(n) \e^{2\pi i n x}/\sqrt{n}$ ($x\in \Q$), a problem for which known approaches based on Eisenstein series have been so far ineffective. We give a new proof, based on dynamical systems, that central modular symbols associated with a holomorphic cusp form for~$SL(2,\Z)$ have a Gaussian distribution, and give the first proof of an estimate for their probabilities of large deviations. We also recover a result of Vardi on the convergence of Dedekind sums to a Cauchy law, using dynamical methods.

Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems

Abstract: The authors develop a theory to compute the statistical properties of time-average observables and find a bound on fluctuations of time-averages from the emergence of the central limit theorem.

Applications of mesoscopic CLTs in random matrix theory

Abstract: We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of “homogenization” for Dyson Brownian motion, this yields the universality of quantities which depend on the behavior of single eigenvalues of Wigner matrices and $\beta$-ensembles. Among the results we obtain are the Gaussian fluctuations of single eigenvalues for Wigner matrices (without an assumption of 4 matching moments) and classical $\beta$-ensembles ($\beta=1,2,4$), Gaussian fluctuations of the eigenvalue counting function, and an asymptotic expansion up to order $o(N^{-1})$ for the expected value of eigenvalues in the bulk of the spectrum. The latter result solves a conjecture of Tao and Vu.

High-dimensional Central Limit Theorems by Stein's Method

Abstract: We obtain explicit error bounds for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random variables or a sum of $n$ locally dependent random vectors. We assume the approximating normal distribution has a non-singular covariance matrix. The error bounds vanish even when the dimension $d$ is much larger than the sample size $n$. We prove our main results using the approach of Gotze (1991) in Stein's method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of $n$ independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a $\log n$ factor. We also discuss an application to multiple Wiener-Ito integrals.

Deviation inequalities for random walks

Abstract: We study random walks on groups, with the feature that, roughly speaking, successive positions of the walk tend to be “aligned.” We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences, including central limit theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In the second part of this article, we show that the (exponential) deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups. These include nonelementary (relatively) hyperbolic groups, mapping class groups, many groups acting on CAT(0) spaces, and small cancellation groups.

Averaging Gaussian functionals

Abstract: This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati’s Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind. The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel $\gamma _{0}$ is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits the Gaussian fluctuation; with some extra mild integrability condition on $\gamma _{0}$, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true.

Limits of random tree-like discrete structures

Abstract: We study a model of random R-enriched trees that is based on weights on the R-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits describing local convergence around fixed and random points in this general context, limit theorems for component sizes when R is a composite class, and a Gromov–Hausdorff scaling limit of random metric spaces patched together from independently drawn metrics on the R-structures. Our main applications treat a selection of examples encompassed by this model. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. We consider random connected graphs drawn according to weights assigned to their blocks and establish a Benjamini–Schramm limit. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest 2-connected component in random graphs from planar-like classes. We prove Benjamini–Schramm convergence of random k-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their 2-connected components.

Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem

Abstract: Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineer...

Nearly optimal central limit theorem and bootstrap approximations in high dimensions

Abstract: In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of $n$ independent high-dimensional centered random vectors $X_1,\dots,X_n$ over the class of rectangles in the case when the covariance matrix of the scaled average is non-degenerate. In the case of bounded $X_i$'s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form $$C (B^2_n \log^3 d/n)^{1/2} \log n,$$ where $d$ is the dimension of the vectors and $B_n$ is a uniform envelope constant on components of $X_i$'s. This bound is sharp in terms of $d$ and $B_n$, and is nearly (up to $\log n$) sharp in terms of the sample size $n$. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded $X_i$'s, formulated solely in terms of moments of $X_i$'s. Finally, we demonstrate that the bounds can be further improved in some special smooth and zero-skewness cases.

Central limit theorems for stochastic wave equations in dimensions one and two

Abstract: Fix $d\in\{1,2\}$, we consider a $d$-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang's condition. In this setting, we provide quantitative central limit theorems for the spatial average of the solution over a Euclidean ball, as the radius of the ball diverges to infinity. We also establish functional central limit theorems. A fundamental ingredient in our analysis is the pointwise $L^p$-estimate for the Malliavin derivative of the solution, which is of independent interest. This paper is another addendum to the recent research line of averaging stochastic partial differential equations.

Two Groups in a Curie–Weiss Model with Heterogeneous Coupling

Abstract: We discuss a Curie–Weiss model with two groups with different coupling constants within and between groups. For the total magnetisations in each group, we show bivariate laws of large numbers and a central limit theorem which is valid in the high-temperature regime. In the critical regime, the total magnetisation normalised by $$N^{3/4}$$ converges to a non-trivial distribution which is not Gaussian, just as in the single-group Curie–Weiss model. Finally, we prove a kind of a ‘law of large numbers’ in the low-temperature regime, more precisely we prove that the empirical magnetisation converges in distribution to a mixture of two Dirac measures.

Functional Limit Theorems for Non-Markovian Epidemic Models

Abstract: We study non-Markovian stochastic epidemic models (SIS, SIR, SIRS, and SEIR), in which the infectious (and latent/exposing, immune) periods have a general distribution. We provide a representation of the evolution dynamics using the time epochs of infection (and latency/exposure, immunity). Taking the limit as the size of the population tends to infinity, we prove both a functional law of large number (FLLN) and a functional central limit theorem (FCLT) for the processes of interest in these models. In the FLLN, the limits are a unique solution to a system of deterministic Volterra integral equations, while in the FCLT, the limit processes are multidimensional Gaussian solutions of linear Volterra stochastic integral equations. In the proof of the FCLT, we provide an important Poisson random measures representation of the diffusion-scaled processes converging to Gaussian components driving the limit process.

Limit Theorems for Random Expanding or Anosov Dynamical Systems and Vector-Valued Observables

Abstract: The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic (Anosov) dynamics developed by the first author et al. to establish quenched versions of the large deviation principle, central limit theorem and the local central limit theorem for vector-valued observables. We stress that the previous works considered exclusively the case of scalar-valued observables. In another direction, we show that this method can be used to establish a variety of new limit laws (either for scalar or vector-valued observables) that have not been discussed previously in the literature for the classes of dynamics we consider. More precisely, we establish the moderate deviations principle, concentration inequalities, Berry–Esseen estimates as well as Edgeworth and large deviation expansions. Although our techniques rely on the approach developed in the previous works of the first author et al., we emphasize that our arguments require several nontrivial adjustments as well as new ideas.

Inference on a distribution function from ranked set samples

Abstract: Consider independent observations $$(X_i,R_i)$$ with random or fixed ranks $$R_i$$, while conditional on $$R_i$$, the random variable $$X_i$$ has the same distribution as the $$R_i$$-th order statistic within a random sample of size k from an unknown distribution function F. Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F: a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment.

Logarithmic Sobolev inequalities for finite spin systems and applications

Abstract: We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity. This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but around a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdős–Renyi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

Testing goodness of fit for point processes via topological data analysis

Abstract: We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

Product and Ratio of Product of Fisher-Snedecor ℱ Variates and Their Applications to Performance Evaluations of Wireless Communication Systems

Abstract: This article considers the product, and the ratio of the product of Fisher-Snedecor $\mathcal {F}$ random variables (RVs), which can be used in modeling fading conditions that are encountered in realistic wireless transmission. To this end, exact analytical expressions are derived for the probability density function (PDF) and cumulative distribution function (CDF) of the product of $N$ statistically independent, but not necessarily identically distributed, Fisher-Snedecor $\mathcal {F}$ RVs. Capitalizing on these, exact analytical expressions are then derived for the outage probability, average channel capacity and average bit error probability over cascaded fading channels. Moreover, some important statistical metrics such as amount of fading, channel quality estimation index, kurtosis, and skewness are also provided, since they provide useful insights on the characteristics of the encountered fading conditions. In addition, with the aid of the central limit theorem, an approximation for the PDF of $N*$ Fisher-Snedecor $\mathcal {F}$ RVs is proposed using a lognormal density, and its accuracy is quantified in terms of the resistor-average distance. Finally, novel expressions for the PDF and CDF of the $N$ -fold product ratio of Fisher-Snedecor $\mathcal {F}$ RVs are also derived. As a potential application of our new results, a spectrum sharing network is considered, for which exact analytical expressions for the outage probability, delay-limited capacity, and ergodic capacity are derived. For the cascaded fading scenario and the spectrum sharing network, numerical examples are provided to show the impact of different channel-related parameters, such as fading severity, shadowing, peak and average interference power on the system performance, which is rather useful in the design of conventional and emerging wireless communication systems. Monte-Carlo simulation results are provided to corroborate the presented mathematical analysis.