Central limit theorem over non-linear functionals of empirical measures: Beyond the iid setting

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

Uniform Estimates of the Rate of Convergence in the Multi-Dimensional Central Limit Theorem

On the Rate of Convergence in the Central Limit Theorem in Certain Banach Spaces

Contents Introduction 1. Formulation of the problem 2. Survey of results 3. Basic results 4. Contents of the article §1. The phase space of a Markov chain 1. The space Y 2. Metric on Y 3. Group action of G on Y 4. Characteristic measures on Y §2. Strong law of large numbers 1. μ-random walk on Y 2. Ergodicity of the μ-random walk 3. Non-equality of Lyapunov exponents 4. Estimates of z(yg(n)) 5. Proof of Theorem 0.1 6. Rate of contraction to a point §3. Limit theorems for Markov chains 1. The Ionescu-Tulcea and Marinescu theorem 2. Perturbed Markov operators 3. Decomposition of Pη(τ) for small τ 4. Central limit theorem 5. Local limit theorem §4. Proof of the central and local limit theorems §5. Proof of the conditional limit theorem 1. Properties of the operator Kβ 2. Properties of the operator Pβ(τ) 3. Critical case 4. Proof of the conditional limit theorem Bibliography

Central and local limit theorems for the coefficients of polynomials of binomial type

In this paper we shall give a brief review of recent results on both the central limit theorems and the non-central limit theorems for non-linear functions of a stationary Gaussian process.

We give criteria for a sequence (X n ) of i.i.d.r.v.'s to satisfy the a.s. central limit theorem, i.e., $$\mathop {\lim }\limits_{N \to \infty } \frac{1}{{\log N}}\sum\limits_{k \leqslant N} {\frac{1}{k}I\left\{ {\frac{{S_k }}{{a_k }} - b_k< x} \right\} = \phi (x)} a.s{\mathbf{ }}for{\mathbf{ }}all{\mathbf{ }}x$$ a.s. for allx whereS n =X 1+X 2+...+X n , (a n ) and (b n ) are numerical sequences andI denotes indicator function. By a recent result of Lacey and Philipp,(7) and Fisher,(6) and its converse obtained here, the a.s. central limit theorem holds with $$a_n = \sqrt n $$ iffEX 1 2 <+∞ which shows that for $$a_n = \sqrt n $$ the a.s. central limit theorem and the corresponding ordinary (weak) central limit theorem are equivalent. Our main results show that for general (a n ) the situation is radically different and paradoxical: the weak central limit theorem implies the a.s. central limit theorem but the converse is false and in fact the validity of the a.s. central limit theorem permits very irregular distributional behavior ofS n /a n −b n . We also show that the validity of the a.s. central limit theorem is closely connected with the limiting behavior of the classical fractionx 2(1−F(x)+F(−x))/∫|t|≤x t 2 dF(t) whereF is the distribution function ofX 1.

It is shown that (to the extent that the moments involved exist) the existence (≠0) of all (generalized) integral scales is necessary (and sufficient if all moments exist) for integrals over adjacent segments of a stationary process to become asymptotically independent, and sufficient to ensure that existing moments of integrals will become Gaussian. The conditions under which several recent central limit and related theorems for dependent variables have been proven, are shown to be closely related to this requirement. As a consequence of this examination, a slight weakening is suggested of the common condition that the spectrum be non-zero. Several physical problems are described, which may be resolved by the application of such a central limit theorem: longitudinal dispersion in a channel flow (previously treated semi-empirically); the spreading of hot spots, or the expansion of macromolecules; the weak interaction hypothesis (of Kraichnan) for Fourier components. Finally, it is shown that dispersion in homogeneous turbulence is unlikely to be explicable on the basis of a central limit theorem.KeywordsCentral Limit TheoremGaussian ProcessAdjacent SegmentIntegral ScaleHomogeneous TurbulenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $\Pr(n^{-1/2}\sum_{i=1}^n X_i\in A)$ where $X_1,\dots,X_n$ are independent random vectors in $\mathbb{R}^p$ and $A$ is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_n\to \infty$ as $n \to \infty$ and $p \gg n$; in particular, $p$ can be as large as $O(e^{Cn^c})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_i$. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $\mathrm{P}(n^{-1/2}\sum_{i=1}^{n}X_{i}\in A)$ where $X_{1},\dots,X_{n}$ are independent random vectors in $\mathbb{R}^{p}$ and $A$ is a hyperrectangle, or more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_{n}\to\infty$ as $n\to\infty$ and $p\gg n$; in particular, $p$ can be as large as $O(e^{Cn^{c}})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_{i}$. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

The [Formula: see text]-bit is the [Formula: see text]-deformation of the [Formula: see text]-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the [Formula: see text]-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of [Formula: see text]-deformation (generalizing the Hudson–Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right [Formula: see text]-JW-embeddings of [Formula: see text] ([Formula: see text] complex matrices) into the infinite tensor product [Formula: see text], proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of [Formula: see text]-Levy processes on ∗-bi-algebras. For [Formula: see text], left [Formula: see text]-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For [Formula: see text], they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all [Formula: see text]. The moments of the limit random variable coincide with those found by Parthasarathy in the case [Formula: see text]. We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case [Formula: see text] and a non-trivial deformation of it for [Formula: see text]. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paper1 deals with the functional extension of this CLT, leading to a process. In paper2 the left [Formula: see text]-JW–embeddings are replaced by symmetric [Formula: see text]-embeddings. The radical differences between the results of the present paper and those of2 raise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paper3 for CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.

Some Limit Theorems for Large Deviations

On the Accuracy of Normal Approximation of the Probability of Hitting a Ball

Previous article Next article Four Examples of Lower Estimates in the Multi-Dimensional Central Limit TheoremV. V. SenatovV. V. Senatovhttps://doi.org/10.1137/1130098PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Senatov, Uniform estimates of the rate of convergence in the multi-dimensional central limit theorem, Theory Prob. Appl., 25 (1980), 745–759 0471.60031 LinkGoogle Scholar[2] V. V. Senatov, On estimating the rate of convergence in the central limit theorem over a system of balls in $R^k$, Theory Prob. Appl., 28 (1983), 463–467 0569.60022 LinkGoogle Scholar[3] L. V. Osipov and , V. I. Rotar', On the convergence rate in the multidimensional and infinite-dimensional central limit theorems, Third International Vilnius Conference on Probability Theory and Mathematical Statistics, Summaries of Reports. V.2, Institute of Mathematics and Cybernetics of the Lithuanian Acad of Sciences, Vilnius, 1981, 97–98, (In Russian.) Google Scholar[4] V. V. Senatov, On the dependence of estimates of the convergence rate in the central limit theorem on the covariance operator of the summands, Theory Prob. Appl., 30 (1985), 156–159 Google Scholar[5] V. V. Yurinskii, On the accuracy of normal approximation of the probability of hitting a ball, Theory Prob. Appl., 27 (1982), 280–289 0565.60005 LinkGoogle Scholar[6] S. V. Nagaev, On accuracy of normal approximation for distribution of sum of independent Hilbert space-valued random variables, 4th USSR-Japan Symposium on Probability Theory and Mathematical Statistics, Abstracts of Communications. V.2, Metsniyereba. Tbilisi, 1982, 130– Google Scholar[7] F. Götze, Asymptotic expansions for bivariate von Mises functionals, Z. Wahrsch. Verw. Gebiete, 50 (1979), 333–355 81c:60025 0405.60009 CrossrefGoogle Scholar[8] V. V. Senatov, Some non-uniform estimates of the speed of convergence in the multi-dimensional central limit theorem, Theory Prob. Appl., 26 (1981), 657–669 0518.60025 LinkGoogle Scholar[9] V. V. Senatov, Some lower bounds for the rate of convergence in the central limit theorem in Hilbert space, Dokl. Akad. Nauk SSSR, 256 (1981), 1318–1322 82e:60011 0474.60007 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Some Approximation Problems in Statistics and Probability9 March 2013 Cross Ref On Estimates of the Rate of Convergence in the Central Limit Theorem in Multidimensional SpacesV. V. Senatov17 July 2006 | Theory of Probability & Its Applications, Vol. 37, No. 4AbstractPDF (259 KB)Some Remarks on Estimating the Rate of Convergence in the Central Limit Theorem in the Hilbert SpaceV. V. Senatov17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 2AbstractPDF (614 KB)Some Remarks on Nonuniform Estimates of the Convergence RateV. V. Senatov17 July 2006 | Theory of Probability & Its Applications, Vol. 33, No. 3AbstractPDF (437 KB)A Regular Estimate of the Accuracy of Normal Approximation in Hilbert SpaceB. A. Zalesskii, V. V. Sazonov, and V. V. Ul’yanov28 July 2006 | Theory of Probability & Its Applications, Vol. 33, No. 4PDF (221 KB)Bibliography Cross Ref Volume 30, Issue 4| 1986Theory of Probability & Its Applications History Submitted:27 October 1983Published online:28 July 2006 InformationCopyright © 1986 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1130098Article page range:pp. 797-805ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

A universal result in almost sure central limit theory