Showing papers on "Central limit theorem published in 2012"

Normal Approximations with Malliavin Calculus

Abstract: Stein's method is a collection of probabilistic techniques that allow one to assess the distance between two probability distributions by means of differential operators. In 2007, the authors discovered that one can combine Stein's method with the powerful Malliavin calculus of variations, in order to deduce quantitative central limit theorems involving functionals of general Gaussian fields. This book provides an ideal introduction both to Stein's method and Malliavin calculus, from the standpoint of normal approximations on a Gaussian space. Many recent developments and applications are studied in detail, for instance: fourth moment theorems on the Wiener chaos, density estimates, Breuer–Major theorems for fractional processes, recursive cumulant computations, optimal rates and universality results for homogeneous sums. Largely self-contained, the book is perfect for self-study. It will appeal to researchers and graduate students in probability and statistics, especially those who wish to understand the connections between Stein's method and Malliavin calculus.

Statistical topological data analysis using persistence landscapes

Abstract: We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries. Viewed as a random variable with values in a Banach space, this summary obeys a strong law of large numbers and a central limit theorem. We show how a number of standard statistical tests can be used for statistical inference using this summary. We also prove that this summary is stable and that it can be used to provide lower bounds for the bottleneck and Wasserstein distances.

Functional Properties of Minimum Mean-Square Error and Mutual Information

Abstract: In addition to exploring its various regularity properties, we show that the minimum mean-square error (MMSE) is a concave functional of the input-output joint distribution. In the case of additive Gaussian noise, the MMSE is shown to be weakly continuous in the input distribution and Lipschitz continuous with respect to the quadratic Wasserstein distance for peak-limited inputs. Regularity properties of mutual information are also obtained. Several applications to information theory and the central limit theorem are discussed.

Bootstrap percolation on the random graph Gn;p

Abstract: Bootstrap percolation on the random graph C-n,C-p is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least r >= 2 active neighbors become active as well. We study the size A* of the final active set. The parameters of the model are, besides r (fixed) and n (tending to infinity), the size a = a(n) of the initially active set and the probability p = p(n) of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either n - o(n) or it is o(n). We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for A*; we also prove a central limit theorem for A* in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.

Random matrices: Universality of local spectral statistics of non-Hermitian matrices

Abstract: It is a classical result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on $\mathbb{C}$ with kernel $K_{\infty}(z,w):=\frac{1}{\pi}e^{-|z|^2/2-|w|^2/2+z\bar{w}}$ in the limit $n\to\infty$. In this paper, we show that this asymptotic law is universal among all random $n\times n$ matrices $M_n$ whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles. These results are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a key new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this issue, we the need to work with the log-determinants $\log|\det(M_n-z_0)|$ rather than with the Stieltjes transform $\frac{1}{n}\operatorname {tr}(M_n-z_0)^{-1}$, in order to exploit Girko's Hermitization method. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real $n\times n$ matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has $\sqrt{\frac{2n}{\pi}}+o(\sqrt{n})$ real eigenvalues asymptotically almost surely.

Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation

Abstract: iffusive phenomena in statistical mechanics and in other fields arise from markovian modeling and their study requires sophisticated mathematical tools. In infinite dimensional situations, time symmetry properties can be exploited in order to make martingale approximations, along the lines of the seminal work of Kipnis and Varadhan. The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior). There are no other books in the mathematical literature that deal with this kind of approach to the problem of the central limit theorem. Hence, this volume meets the demand for a monograph on this powerful approach, now widely used in many areas of probability and mathematical physics. The book also covers the connections with and application to hydrodynamic limits and homogenization theory, so besides probability researchers it will also be of interest to mathematical physicists and analysts.

An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums

Abstract: By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F n ,Φ) between the standard normal distribution function Φ and the distribution function F n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β3. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β3+0.429)≤0.33477(1+0.429)β3<0.4784β3 by virtue of the condition β3≥1. The latter inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry–Esseen inequality for Poisson random sums to 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequalit...

Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring

Abstract: In this paper we address the following question: "Can we approximately sample from a Bayesian posterior distribution if we are only allowed to touch a small mini-batch of data-items for every sample we generate?". An algorithm based on the Langevin equation with stochastic gradients (SGLD) was previously proposed to solve this, but its mixing rate was slow. By leveraging the Bayesian Central Limit Theorem, we extend the SGLD algorithm so that at high mixing rates it will sample from a normal approximation of the posterior, while for slow mixing rates it will mimic the behavior of SGLD with a pre-conditioner matrix. As a bonus, the proposed algorithm is reminiscent of Fisher scoring (with stochastic gradients) and as such an efficient optimizer during burn-in.

Gaussian and Non-Gaussian Linear Time Series and Random Fields

Abstract: 1 Reversibility and Identifiability.- 1.1 Linear Sequences and the Gaussian Property.- 1.2 Reversibility.- 1.3 Identifiability.- 1.4 Minimum and Nonminimum Phase Sequences.- 2 Minimum Phase Estimation.- 2.1 The Minimum Phase Case and the Quasi-Gaussian Likelihood.- 2.2 Consistency.- 2.3 The Asymptotic Distribution.- 3 Homogeneous Gaussian Random Fields.- 3.1 Regular and Singular Fields.- 3.2 An Isometry.- 3.3 L-Fields and L-Markov Fields.- 4 Cumulants, Mixing and Estimation for Gaussian Fields.- 4.1 Moments and Cumulants.- 4.2 Higher Order Spectra.- 4.3 Some Simple Inequalities and Strong Mixing.- 4.4 Strong Mixing for Two-Sided Linear Processes.- 4.5 Mixing and a Central Limit Theorem for Random Fields.- 4.6 Estimation for Stationary Random Fields.- 4.7 Cumulants of Finite Fourier Transforms.- 4.8 Appendix: Two Inequalities.- 5 Prediction for Minimum and Nonminimum Phase Models.- 5.1 Introduction.- 5.2 A First Order Autoregressive Model.- 5.3 Nonminimum Phase Autoregressive Models.- 5.4 A Functional Equation.- 5.5 Entropy.- 5.6 Continuous Time Parameter Processes.- 6 The Fluctuation of the Quasi-Gaussian Likelihood.- 6.1 Initial Remarks.- 6.2 Derivation.- 6.3 The Limiting Process.- 7 Random Fields.- 7.1 Introduction.- 7.2 Markov Fields and Chains.- 7.3 Entropy and a Limit Theorem.- 7.4 Some Illustrations.- 8 Estimation for Possibly Nonminimum Phase Schemes.- 8.1 The Likelihood for Possibly Non-Gaussian Autoregressive Schemes.- 8.2 Asymptotic Normality.- 8.3 Preliminary Comments: Approximate Maximum Likelihood Estimates for Non-Gaussian Nonminimum Phase ARMA Sequences.- 8.4 The Likelihood Function.- 8.5 The Covariance Matrix.- 8.6 Solution of the Approximate Likelihood Equations.- 8.7 Cumulants and Estimation for Autoregressive Schemes.- 8.8 Superefficiency.- Bibliographic Notes.- References.- Notation.- Author Index.

Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring

Abstract: In this paper we address the following question: Can we approximately sample from a Bayesian posterior distribution if we are only allowed to touch a small mini-batch of data-items for every sample we generate?. An algorithm based on the Langevin equation with stochastic gradients (SGLD) was previously proposed to solve this, but its mixing rate was slow. By leveraging the Bayesian Central Limit Theorem, we extend the SGLD algorithm so that at high mixing rates it will sample from a normal approximation of the posterior, while for slow mixing rates it will mimic the behavior of SGLD with a pre-conditioner matrix. As a bonus, the proposed algorithm is reminiscent of Fisher scoring (with stochastic gradients) and as such an efficient optimizer during burn-in.

Performance of a Distributed Stochastic Approximation Algorithm

Abstract: In this paper, a distributed stochastic approximation algorithm is studied. Applications of such algorithms include decentralized estimation, optimization, control or computing. The algorithm consists in two steps: a local step, where each node in a network updates a local estimate using a stochastic approximation algorithm with decreasing step size, and a gossip step, where a node computes a local weighted average between its estimates and those of its neighbors. Convergence of the estimates toward a consensus is established under weak assumptions. The approach relies on two main ingredients: the existence of a Lyapunov function for the mean field in the agreement subspace, and a contraction property of the random matrices of weights in the subspace orthogonal to the agreement subspace. A second order analysis of the algorithm is also performed under the form of a Central Limit Theorem. The Polyak-averaged version of the algorithm is also considered.

Wigner chaos and the fourth moment

Abstract: We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer–Major theorem.

Quarticity and other functionals of volatility: Efficient estimation

Abstract: We consider a multidimensional Ito semimartingale regularly sampled on [0,t] at high frequency $1/\Delta_n$, with $\Delta_n$ going to zero. The goal of this paper is to provide an estimator for the integral over [0,t] of a given function of the volatility matrix. To approximate the integral, we simply use a Riemann sum based on local estimators of the pointwise volatility. We show that although the accuracy of the pointwise estimation is at most $\Delta_n^{1/4}$, this procedure reaches the parametric rate $\Delta_n^{1/2}$, as it is usually the case in integrated functionals estimation. After a suitable bias correction, we obtain an unbiased central limit theorem for our estimator and show that it is asymptotically efficient within some classes of sub models.

Scaling limits for Hawkes processes and application to financial statistics

Abstract: We prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval $[0,T]$ in the limit $T \rightarrow \infty$. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh $\Delta$ over $[0,T]$ up to some further time shift $\tau$. The behaviour of this functional depends on the relative size of $\Delta$ and $\tau$ with respect to $T$ and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in a previous work a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce important empirical stylised fact such as the Epps effect and the lead-lag effect. Moreover, our approach enable to track these effects across scales in rigorous mathematical terms.

Sticky central limit theorems on open books

Abstract: Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fr\'{e}chet mean (barycenter) is sticky This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension $1$ and hence measure $0$) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine We also state versions of the LLN and CLT for the cases where the mean is nonsticky (ie, not lying on the spine) and partly sticky (ie, is, on the spine but not sticky)

Random matrices: The Universality phenomenon for Wigner ensembles

Abstract: In this paper, we survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Central limit theorem of several spectral parameters. We also take the opportunity here to issue some errata for some of our previous papers in this area.

Identifying the brownian covariation from the co-jumps given discrete observations

Abstract: When the covariance between the risk factors of asset prices is due to both Brownian and jump components, the realized covariation (RC) approaches the sum of the integrated covariation (IC) with the sum of the co-jumps, as the observation frequency increases to infinity, in a finite and fixed time horizon. In this paper the two components are consistently separately estimated within a semimartingale framework with possibly infinite activity jumps. The threshold (or truncated) estimator is used, which substantially excludes from RC all terms containing jumps. Unlike in Jacod (2007, Universite de Paris-6) and Jacod (2008, Stochastic Processes and Their Applications 118, 517–559), no assumptions on the volatilities’ dynamics are required. In the presence of only finite activity jumps: 1) central limit theorems (CLTs) for and for further measures of dependence between the two Brownian parts are obtained; the estimation error asymptotic variance is shown to be smaller than for the alternative estimators of IC in the literature; 2) by also selecting the observations as in Hayashi and Yoshida (2005, Bernoulli 11, 359–379), robustness to nonsynchronous data is obtained. The proposed estimators are shown to have good finite sample performances in Monte Carlo simulations even with an observation frequency low enough to make microstructure noises’ impact on data negligible.

A student's guide to data and error analysis

Abstract: Part I. Data and Error Analysis: 1. Introduction 2. The presentation of physical quantities with their inaccuracies 3. Errors: classification and propagation 4. Probability distributions 5. Processing of experimental data 6. Graphical handling of data with errors 7. Fitting functions to data 8. Back to Bayes: knowledge as a probability distribution Answers to exercises Part II. Appendices: A1. Combining uncertainties A2. Systematic deviations due to random errors A3. Characteristic function A4. From binomial to normal distributions A5. Central limit theorem A6. Estimation of the varience A7. Standard deviation of the mean A8. Weight factors when variances are not equal A9. Least squares fitting Part III. Python Codes Part IV. Scientific Data: Chi-squared distribution F-distribution Normal distribution Physical constants Probability distributions Student's t-distribution Units.

Moderate Deviations in Channel Coding

Abstract: We consider block codes whose rate converges to the channel capacity with increasing block length at a certain speed and examine the best possible decay of the probability of error. We prove that a moderate deviation principle holds for all convergence rates between the large deviation and the central limit theorem regimes.

Estimation of Nonlinear Functionals of Densities With Confidence

Abstract: This paper introduces a class of k-nearest neighbor (k-NN) estimators called bipartite plug-in (BPI) estimators for estimating integrals of nonlinear functions of a probability density, such as Shannon entropy and Renyi entropy. The density is assumed to be smooth, have bounded support, and be uniformly bounded from below on this set. Unlike previous k-NN estimators of nonlinear density functionals, the proposed estimator uses data-splitting and boundary correction to achieve lower mean square error. Specifically, we assume that T i.i.d. samples Xi ϵ Rd from the density are split into two pieces of cardinality M and N, respectively, with M samples used for computing a k-NN density estimate and the remaining N samples used for empirical estimation of the integral of the density functional. By studying the statistical properties of k-NN balls, explicit rates for the bias and variance of the BPI estimator are derived in terms of the sample size, the dimension of the samples, and the underlying probability distribution. Based on these results, it is possible to specify optimal choice of tuning parameters M/T, k for maximizing the rate of decrease of the mean square error. The resultant optimized BPI estimator converges faster and achieves lower mean squared error than previous k-NN entropy estimators. In addition, a central limit theorem is established for the BPI estimator that allows us to specify tight asymptotic confidence intervals.

Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models

Abstract: We present limit theorems for a sequence of Piecewise Deterministic Markov Processes (PDMPs) taking values in a separable Hilbert space. This class of processes provides a rigorous framework for stochastic spatial models in which discrete random events are globally coupled with continuous space dependent variables solving partial differential equations, e.g., stochastic hybrid models of excitable membranes. We derive a law of large numbers which establishes a connection to deterministic macroscopic models and a martingale central limit theorem which connects the stochastic fluctuations to diffusion processes. As a prerequisite we carry out a thorough discussion of Hilbert space valued martingales associated to the PDMPs. Furthermore, these limit theorems provide the basis for a general Langevin approximation to PDMPs, i.e., stochastic partial differential equations that are expected to be similar in their dynamics to PDMPs. We apply these results to compartmental-type models of spatially extended excitable membranes. Ultimately this yields a system of stochastic partial differential equations which models the internal noise of a biological excitable membrane based on a theoretical derivation from exact stochastic hybrid models.

Fractional governing equations for coupled random walks

Abstract: In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW is coupled if the waiting time and the subsequent jump are dependent random variables. The CTRW is used in physics to model diffusing particles. Its scaling limit is governed by an anomalous diffusion equation. Some applications require an overshoot continuous time random walk (OCTRW), where the waiting time is coupled to the previous jump. This paper develops stochastic limit theory and governing equations for CTRW and OCTRW. The governing equations involve coupled space-time fractional derivatives. In the case of infinite mean waiting times, the solutions to the CTRW and OCTRW governing equations can be quite different.

Universality for first passage percolation on sparse random graphs

Abstract: We consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform X^2\log{X}-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path or hopcount. The hopcount satisfies a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous-time branching process. Centered by a multiple of \log{n}, where the constant is the inverse of the Malthusian rate of growth of the associated branching process, the minimal weight converges in distribution. The limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices, and a Gumbel random variable, and thus shows a remarkably universal behavior. The proofs rely on a refined coupling between the shortest path problems on these graphs and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination. The results extend to a host of related random graph models, ranging from random r-regular graphs, inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence.

Central limit theorems for the excursion set volumes of weakly dependent random fields

Abstract: The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on ℝd are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the covariance matrix of the limiting distribution are provided. A statistical version of the CLT is considered as well. Some numerical results are also discussed.

The averaged periodogram estimator for a power law in coherency

Abstract: We prove the consistency of the averaged periodogram estimator (APE) in two new cases. First, we prove that the APE is consistent for negative memory parameters, after suitable tapering. Second, we prove that the APE is consistent for a power law in the cross-spectrum and therefore for a power law in the coherency, provided that su-ciently many frequencies are used in estimation. Simulation evidence suggests that the lower bound on the number of frequencies is a necessary condition for consistency. For a Taylor series approximation to the estimator of the power law in the cross-spectrum, we consider the rate of convergence, and obtain a central limit theorem under suitable regularity conditions.