Showing papers on "Central limit theorem published in 2022"
TL;DR: In this article, simultaneous confidence bands (SCBs) for functional parameters over arbitrary dimensional compact domains using the Gaussian Kinematic formula of t -processes (tGKF) were proposed.
16 citations
14 citations
TL;DR: In this article , a sharp uniform-in-bandwidth limit law for nonparametric estimation of a parameter, which is a zero of a certain estimating equation, indexed by a class of functions and depending on an infinite-dimensional covariate, is presented.
12 citations
TL;DR: In this paper , simultaneous confidence bands (SCBs) for functional parameters over arbitrary dimensional compact domains using the Gaussian Kinematic formula of t-processes (tGKF) were proposed.
11 citations
TL;DR: It is proved that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix has approximately Gaussian fluctuation for the bulk spectrum.
Abstract: Abstract We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.
8 citations
TL;DR: In this paper , Huang, Nualart, and Viitasaari used the Malliavin-Stein method to prove that the random field x ↦ u (t, x ) / p t (x ) is ergodic for every t > 0.
8 citations
TL;DR: In this article , the authors used the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic, for generic $\beta$-ensembles.
Abstract: In the setting of generic $\beta$-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of order $(\log N)/N$ in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale $\sqrt{\log N}/N$. (iii) The logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem. Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian $\beta$-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.
8 citations
TL;DR: In this paper , the authors studied the limiting distribution 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.
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 Vallée 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 ∑ n ⩾ 1 τ ( n ) e 2 π i n x / n $\sum _{n\geqslant 1} \tau (n) {\rm e}^{2\pi i n x}/\sqrt {n}$ ( x ∈ Q $x\in {\mathbb {Q}}$ ), 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 S L ( 2 , Z ) $SL(2,{\mathbb {Z}})$ 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.
7 citations
TL;DR: In this article , the authors studied limit theorems for time-dependent averages of the form Xt: = 12L(t)∫−L (t) ∫ −L(T)L( t)L (T)u(t,x)dx, as t→∞, where L(t)=exp(λt) and u(t),x) is the solution to a stochastic heat equation on ℝ +×ℝ driven by space-time white noise with u0(x)=1 for all x∈℞.
Abstract: We study limit theorems for time-dependent averages of the form Xt:=12L(t)∫−L(t)L(t)u(t,x)dx, as t→∞, where L(t)=exp(λt) and u(t,x) is the solution to a stochastic heat equation on ℝ +×ℝ driven by space-time white noise with u0(x)=1 for all x∈ℝ. We show that for Xt the weak law of large numbers holds when λ>λ1, the strong law of large numbers holds when λ>λ2, the central limit theorem holds when λ>λ3, but fails when λ<λ4≤λ3, the quantitative central limit theorem holds when λ>λ5, where λi’s are positive constants depending on the moment Lyapunov exponents of u(t,x).
7 citations
TL;DR: In this article , it was shown that if a mixing condition holds, the averages also satisfy a central limit theorem, a Berry-Esseen bound, and concentration, and these estimators are strongly consistent.
Abstract: A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory provides a law of large numbers: If the group satisfies suitable conditions, expectations can be estimated by averaging over subsets of transformations, and these estimators are strongly consistent. We show that, if a mixing condition holds, the averages also satisfy a central limit theorem, a Berry-Esseen bound, and concentration. These are extended further to apply to triangular arrays, to randomly subsampled averages, and to a generalization of U-statistics. As applications, we obtain new results on exchangeability, random fields, network models, and a class of marked point processes. We also establish asymptotic normality of the empirical entropy for a large class of processes. Some known results are recovered as special cases, and can hence be interpreted as an outcome of symmetry. The proofs adapt Stein’s method.
7 citations
TL;DR: In this article, Huang, Nualart, and Viitasaari used the Malliavin-Stein method to prove that the random field x ↦ u (t, x ) / p t (x ) is ergodic for every t > 0.
TL;DR: This paper conducted a survey to examine the misconceptions of the central limit theorem among graduate students and researchers in the social sciences and found that the most common misconception is that researchers think the CLT is about the convergence of sample data to the normal distribution.
Abstract: The central limit theorem (CLT) is one of the most important theorems in statistics, and it is often introduced to social sciences researchers in an introductory statistics course. However, the recent replication crisis in the social sciences prompts us to investigate just how common certain misconceptions of statistical concepts are. The main purposes of this article are to investigate the misconceptions of the CLT among social sciences researchers and to address these misconceptions by clarifying the definition and properties of the CLT in a manner that is approachable to social science researchers. As part of our article, we conducted a survey to examine the misconceptions of the CLT among graduate students and researchers in the social sciences. We found that the most common misconception of the CLT is that researchers think the CLT is about the convergence of sample data to the normal distribution. We also found that most researchers did not realize that the CLT applies to both sample means and sample sums, and that the CLT has implications for many common statistical concepts and techniques. Our article addresses these misconceptions of the CLT by explaining the preliminaries needed to understand the CLT, introducing the formal definition of the CLT, and elaborating on the implications of the CLT. We hope that through this article, researchers can obtain a more accurate and nuanced understanding of how the CLT operates as well as its role in a variety of statistical concepts and techniques. (PsycInfo Database Record (c) 2022 APA, all rights reserved).
TL;DR: In this paper , the authors studied the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension d ≥ 2, as the domain of the integral gets large.
TL;DR: In this paper , an approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general L\'evy process at an arbitrary finite time is presented.
Abstract: We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general L\'evy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in $L^p$ (for any $p\geq 1$) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct non-asymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We prove that the multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $\epsilon^{-2}$ if the mean squared error is at most $\epsilon^2$) for locally Lipschitz and barrier-type functionals of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.
TL;DR: In this article, the authors studied limit theorems for time-dependent averages of the form Xt: = 12L(t)∫−L (t) ∫−∞ L(t), L (t,x) u(t, x) is the solution to a stochastic heat equation on ℝ +×ℝ driven by space-time white noise with u0(x)=1 for all x∈℞.
Abstract: We study limit theorems for time-dependent averages of the form Xt:=12L(t)∫−L(t)L(t)u(t,x)dx, as t→∞, where L(t)=exp(λt) and u(t,x) is the solution to a stochastic heat equation on ℝ +×ℝ driven by space-time white noise with u0(x)=1 for all x∈ℝ. We show that for Xt the weak law of large numbers holds when λ>λ1, the strong law of large numbers holds when λ>λ2, the central limit theorem holds when λ>λ3, but fails when λ λ5, where λi’s are positive constants depending on the moment Lyapunov exponents of u(t,x).
TL;DR: In this article , a central limit theorem for additive functionals of BMCs under three different regimes with three different regime assumptions is provided, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs.
Abstract: Abstract Bifurcating Markov chains (BMCs) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMCs under
$L^2$
-ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As an application, we study the elementary case of a symmetric bifurcating autoregressive process, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs. We illustrate in this example the phase transition observed in the fluctuations.
TL;DR: In this article , the Gaussian and bootstrap approximations to the distribution of the max statistic in high dimensions were studied and new bounds for the distributional approximation errors were derived using a novel iterative randomized Lindeberg method.
Abstract: This paper deals with the Gaussian and bootstrap approximations to the distribution of the max statistic in high dimensions. This statistic takes the form of the maximum over components of the sum of independent random vectors and its distribution plays a key role in many high-dimensional estimation and testing problems. Using a novel iterative randomized Lindeberg method, the paper derives new bounds for the distributional approximation errors. These new bounds substantially improve upon existing ones and simultaneously allow for a larger class of bootstrap methods.
TL;DR: In this article , a functional law of large number (FLLN) and a functional central limit theorem (FCLT) were proved for non-Markovian stochastic epidemic models, in which the infectious (and latent/exposing, immune) periods have a general distribution.
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.
TL;DR: In this paper , the authors investigated the uniform limit theory for a U-statistic of increasing degree, also called an infinite-degree U-Statistic, and obtained the first known results on the infinite degree U-process in the Markovian setting.
TL;DR: In this paper , the authors established a feasible central limit theorem with convergence rate n 1/8 for the estimation of the integrated volatility of volatility (VoV) based on noisy high-frequency data with jumps.
TL;DR: In this paper , the authors investigated the spectral properties of a high-dimensional sample spatial-sign covariance matrix when both the dimension of the observations and the sample size grows to infinity.
Abstract: This paper investigates limiting spectral properties of a high-dimensional sample spatial-sign covariance matrix when both the dimension of the observations and the sample size grow to infinity. The underlying population is general enough to include the popular independent components model and the family of elliptical distributions. The first result of the paper shows that the empirical spectral distribution of a high dimensional sample spatial-sign covariance matrix converges to a generalized Marčenko-Pastur distribution. Secondly, a new central limit theorem for a class of related linear spectral statistics is established.
TL;DR: In this paper , the authors derive properties of the Fréchet mean on S∞ by establishing its existence and uniqueness as well as a root-n central limit theorem (CLT) for the sample version, overcoming obstructions from infinite dimensionality and lack of compactness on S ∞.
Abstract: The infinite-dimensional Hilbert sphere S∞ has been widely employed to model density functions and shapes, extending the finite-dimensional counterpart. We consider the Fréchet mean as an intrinsic summary of the central tendency of data lying on S∞. For sound statistical inference, we derive properties of the Fréchet mean on S∞ by establishing its existence and uniqueness as well as a root-n central limit theorem (CLT) for the sample version, overcoming obstructions from infinite-dimensionality and lack of compactness on S∞. Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Fréchet mean based on projection and norm are then proposed and are shown to be consistent. The proposed two-sample tests are applied to make inference for daily taxi demand patterns over Manhattan, modeled as densities, of which the square root densities are analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis tests which utilize the spherical geometry are studied in the real data application and simulations, where we demonstrate that the tests based on the intrinsic geometry compare favorably to those based on an extrinsic or flat geometry.
TL;DR: A new method via p -Wasserstein bounds is used to prove Cram´er-type moderate deviations in (multivariate) normal approximations and gives applications to the combinatorial central limit theorem, Wiener chaos, homogeneous sums and local dependence.
Abstract: We use a new method via $p$-Wasserstein bounds to prove Cram\'er-type moderate deviations in (multivariate) normal approximations. In the classical setting that $W$ is a standardized sum of $n$ independent and identically distributed (i.i.d.) random variables with sub-exponential tails, our method recovers the optimal range of $0\leq x=o(n^{1/6})$ and the near optimal error rate $O(1)(1+x)(\log n+x^2)/\sqrt{n}$ for $P(W>x)/(1-\Phi(x))\to 1$, where $\Phi$ is the standard normal distribution function. Our method also works for dependent random variables (vectors) and we give applications to the combinatorial central limit theorem, Wiener chaos, homogeneous sums and local dependence. The key step of our method is to show that the $p$-Wasserstein distance between the distribution of the random variable (vector) of interest and a normal distribution grows like $O(p^\alpha \Delta)$, $1\leq p\leq p_0$, for some constants $\alpha, \Delta$ and $p_0$. In the above i.i.d. setting, $\alpha=1, \Delta=1/\sqrt{n}, p_0=n^{1/3}$. For this purpose, we obtain general $p$-Wasserstein bounds in (multivariate) normal approximations using Stein's method.
TL;DR: In this paper , Nualart-Zheng et al. studied the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in both time and space.
Abstract: In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in both time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use Ito calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-order Gaussian Poincaré inequality to overcome this lack of martingale structure and our work improves the results by Nualart-Zheng (Electron. J. Probab. 2020) and Nualart-Song-Zheng (ALEA, Lat. Am. J. Probab. Math. Stat. 2021).
TL;DR: In this paper , the central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way is proved for a continuous-time random walk.
TL;DR: In this article , the authors obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables, including products of polynomials.
TL;DR: In this article , a consistent and asymptotically mixed normal estimator of the Hurst parameter was derived based on high-frequency price observations, and the estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.
Abstract: In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter H < 0 . 5 . In this paper, we derive a consistent and asymptotically mixed normal estimator of H based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.
TL;DR: In this paper , a stochastic compartmental susceptible-infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval is studied.
Abstract: Abstract We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.
TL;DR: In this paper , the central limit theorem for linear processes generated by m-dependent random variables under sub-linear expectations was proved for the case of linear processes with sublinear expectation space.
Abstract: Abstract We prove the Rosnethal’s inequality of m-dependent random variables under the sub-linear expectation in this paper. Furthermore, we use this inequality to investigate the central limit theorem for linear processes generated by m-dependent random variables under sub-linear expectations. This article use the basic definitions of sub-linear expectation space, Kronecker lemma, Cr inequality etc. to demonstrate the main conclusion.