Convergence In Central Limit Theorem Research Articles

Applications of the CLT for Positively Associated Random Process in Time Series Analysis

This paper explores the implications of the Central Limit Theorem (CLT) within the framework of positively associated stationary random fields, which are pivotal in mathematical statistics, reliability theory, percolation, and statistical physics. It delves into the challenges of extending the CLT's convergence rate to these complex fields, building upon the foundational work by Newman and subsequent contributions. The study presents a novel approach to constructing ARMA models that align with the CLT, offering a robust framework for time series analysis. The paper concludes with the significance of these findings for statistical modeling and forecasting in various disciplines.

On the Rates of Convergence in Central Limit Theorems for Compound Random Sums of Independent Random Variables

On the Rates of Convergence in Central Limit Theorems for Compound Random Sums of Independent Random Variables

Оцiнка Беррi-Ессеєна для майже критичних гiллястих процесiв з iммiграцiєю

In this paper, we consider a nearly critical branching process with immigration. We obtain the rate of convergence in central limit theorem for nearly critical branching processes with immigration.

Exact rates of convergence in some martingale central limit theorems

Renz [14], Ouchti [13], El Machkouri and Ouchti [3] and Mourrat [12] have established some tight bounds on the rate of convergence in the central limit theorem for martingales. In the present paper a modification of the methods, developed by Bolthausen [1] and Grama and Haeusler [7], is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order 2+ρ,ρ>0. Our results generalise and strengthen the bounds mentioned above.

Optimal rates for parameter estimation of stationary Gaussian processes

We study rates of convergence in central limit theorems for partial sums of polynomial functionals of general stationary and asymptotically stationary Gaussian sequences, using tools from analysis on Wiener space. In the quadratic case, thanks to newly developed optimal tools, we derive sharp results, i.e. upper and lower bounds of the same order, where the convergence rates are given explicitly in the Wasserstein distance via an analysis of the functionals’ absolute third moments. These results are tailored to the question of parameter estimation, which introduces a need to control variance convergence rates. We apply our result to study drift parameter estimation problems for some stochastic differential equations driven by fractional Brownian motion with fixed-time-step observations.

Berry–Esseen estimates for regenerative processes under weak moment assumptions

We prove Berry–Esseen type rates of convergence for central limit theorems (CLTs) of regenerative processes which generalize previous results of Bolthausen under weaker moment assumptions. We then show how this general result can be applied to obtain rates of convergence for (1) CLTs for additive functionals of positive recurrent Markov chains under certain conditions on the strong mixing coefficients, and (2) annealed CLTs for certain ballistic random walks in random environments.

Taking into account the rate of convergence in CLT under Risk evaluation on financial markets

This paper examines “fat tails puzzle” in the financial markets. Ignoring the rate of convergence in Central Limit Theorem (CLT) provides the “fat tail” uncertainty. In this paper, we provide a rev...

On the rate of convergence in the martingale central limit theorem

Consider a discrete-time martingale, and let $V^2$ be its normalized quadratic variation. As $V^2$ approaches 1, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any $p\geq 1$, (Ann. Probab. 16 (1988) 275-299) gave a bound on the rate of convergence in this central limit theorem that is the sum of two terms, say $A_p+B_p$, where up to a constant, $A_p={\|V^2-1\|}_p^{p/(2p+1)}$. Here we discuss the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, (Ann. Probab. 10 (1982) 672-688) sketched a strategy to prove optimality for $p=1$. Here we extend this strategy to any $p\geq 1$, thereby justifying the optimality of the term $A_p$. As a necessary step, we also provide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that improves on the term $B_p$, generalizing another result of (Ann. Probab. 10 (1982) 672-688).

On the Question of the Fat Tails in Financial Markets

In this paper we constructed bounds for tails of financial distributions using known inequalities for sums of random variables and Central Limit Theorem (CLT); the obtained bounds proved to be quite consistent with the empirical data.The “fat tail” confusion typically occurs when normal distribution is used ignoring the rate of convergence in CLT. For example, historically, annual equity returns experience six-sigma losses with probability around 0.27%, which is 30 million times “fatter” than the corresponding six-sigma normal tail of 0.000000001 (one-billionth). However, this extraordinary “fatness” is explained by a simple observation that the number of days in one year is not sufficient to apply CLT with the one-billionth precision. Our bound, constructed for the same case of annual six-sigma losses, comes around 1%; hence the observed 0.27% tail looks rather fit than fat. The proposed methodology and bounds are based on well-established mathematical results, are practical and easy to use; they may be employed by prudent investors and risk managers to carefully evaluate risks.

On the rate of convergence for central limit theorems of sojourn times of Gaussian fields

The aim of this paper is to control the rate of convergence for central limit theorems of sojourn times of Gaussian fields in both cases: the fixed and the moving level. Our main tools are the Malliavin calculus and the Stein method, developed by Nualart, Peccati and Nourdin. We also extend some results of Berman to the multidimensional case.

Berry-Esseen-type bounds for the kernel estimator of conditional distribution and conditional quantiles

Berry-Esseen bounds of order O(n − 1 2 ) have been obtained for several classes of statistics. In this paper, the rates of convergence in central limit theorem for conditional empirical functions and conditional sample quantiles based on kernel estimators are studied for both conditional and unconditional distributions.

Four Examples of Lower Estimates in the Multi-Dimensional Central Limit Theorem

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

On Estimating the Rate of Convergence in the Central Limit Theorem Over a System of Balls in $R^k $

Previous article Next article On Estimating the Rate of Convergence in the Central Limit Theorem Over a System of Balls in $R^k $V. V. SenatovV. V. Senatovhttps://doi.org/10.1137/1128042PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. M. Zolotarev, Metric distances in spaces of random variables and their distributions, Math. USSR, 30 (1976), 373–401 0383.60022 CrossrefGoogle Scholar[2] 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[3] L. V. Osipov and , V. I. Rotar', On the rate of convergence in the mufti- and infinite-dimensional central limit theorems, Third International Vil'nyus Conference in Probability Theory and Mathematical Statistics, Summary of Reports. Vol. 2,, Inst. Matematiki i Kibernetiki AN Akad. Nauk. LitSSR, Vil'nyus, 1981, 97–98, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Four Examples of Lower Estimates in the Multi-Dimensional Central Limit TheoremV. V. SenatovTheory of Probability & Its Applications, Vol. 30, No. 4 | 28 July 2006AbstractPDF (828 KB)BibliographyNormal Approximation Cross Ref Volume 28, Issue 2| 1984Theory of Probability & Its Applications219-469 History Submitted:17 December 1982Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1128042Article page range:pp. 463-467ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Nonclassical Estimates of the Rate of Convergence in the Central Limit Theorem Taking into Account Large Deviations

Previous article Next article Nonclassical Estimates of the Rate of Convergence in the Central Limit Theorem Taking into Account Large DeviationsS. Ya. ShorginS. Ya. Shorginhttps://doi.org/10.1137/1127034PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Sazonov, A new general estimate of the rate of convergence in the central limit theorem in $R^{k}$, Proc. Nat. Acad. Sci. USA, 71 (1974), 118–121 0277.60010 CrossrefGoogle Scholar[2] V. I. Rotar', Non-classical estimates of the rate of convergence in the multi-dimensional central limit theorem. I, Theory Prob. Appl., 22 (1977), 755–772 0391.60026 LinkGoogle Scholar[3] V. I. Rotar', Non-classical estimates for the rate of convergence in the multidimensional central limit theorem. II, Theory Prob. Appl., 23 (1978), 50–62 0423.60021 LinkGoogle Scholar[4] S. V. Nagaev and , V. I. Rotar', Sharpening of Lyapunov type estimates (the case when the distributions of the summands are close to the normal distribution), Theory Prob. Appl., 18 (1973), 107–119 LinkGoogle Scholar[5] S. Ya. Shorgin, A non-classical estimate of the rate of convergence in the multi-dimensional central limit theorem taking into account large deviations, Theory Prob. Appl., 23 (1978), 667–671 LinkGoogle Scholar[6] S. V. Nagaev and , S. K. Sakoyan, An estimate for the probability of large deviations, Limit theorems and mathematical statistics (Russian), Izdat. “Fan” Uzbek. SSR, Tashkent, 1976, 132–140, 190 55:9237 Google Scholar[7] S. K. Sakoyan, Estimates for distribution functions of sums of random variables taking into account large deviations, 1977, author's abstract of dissertation, Tashkent Google Scholar[8] V. I. Rotar', Masters Thesis, Limit Theorems for Linear and Multilinear Forms, dissertation, LGU, Moscow-Leningrad, 1978 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Limit Theorems on Large DeviationsLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Probabilistic Inequalities for Sums of Independent RandomVariables in Terms of Truncated PseudomomentsS. V. NagaevTheory of Probability & Its Applications, Vol. 42, No. 3 | 17 February 2012AbstractPDF (265 KB)On the Exit of a Random Walk From a Curvilinear BoundaryM. U. Gafurov and V. I. Rotar’Theory of Probability & Its Applications, Vol. 28, No. 1 | 17 July 2006AbstractPDF (477 KB) Volume 27, Issue 2| 1983Theory of Probability & Its Applications215-440 History Submitted:26 March 1980Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1127034Article page range:pp. 324-337ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

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

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

A note on rates of convergence in the multidimensional CLT for maximum partial sums

In this note we obtain rates of convergence in the central limit theorem for certain maximum of coordinate partial sums of independent identically distributed random vectors having positive mean vector and a nonsingular correlation matrix. The results obtained are in terms of rates of convergence in the multidimensional central limit theorem. Thus under the conditions of Sazonov (1968, Sankhya, Series A 30 181–204, Theorem 2), we have the same rate of convergence for the vector of coordinate maximums. Other conditions for the multidimensional CLT are also discussed, c.f., Bhattachaya (1977, Ann. Probability 5 1–27). As an application of one of the results we obtain a multivariate extension of a theorem of Rogozin (1966, Theor. Probability Appl. 11 438–441).

Non-Classical Estimates of the Rate of Convergence in the Multi-Dimensional Central Limit Theorem. II

Previous article Next article Non-Classical Estimates of the Rate of Convergence in the Multi-Dimensional Central Limit Theorem. IIV. I. Rotar’V. I. Rotar’https://doi.org/10.1137/1123004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. I. Rotar', Non-classical estimates of the rate of convergence in the multidimensional central limit theorem. I, Theory Prob. Applications, 22 (1977), 665–682 0391.60026 Google Scholar[2] S. V. Nagaev and , V. I. Rotar', On refinement of Lyapunov type estimates (the case of near to normal distributions of terms), Theory Prob. Applications, 18 (1973), 107–119 10.1137/1118008 0284.60016 LinkGoogle Scholar[3] S. V. Nagaev and , V. I. Rotar', Letter to the editors, Theory Prob. Applications, 21 (1976), 220– LinkGoogle Scholar[4] V. I. Rotar', Non-uniform estimate of the rate of convergence in the multi-dimensional central limit theorem, Theory Prob. Applications, 15 (1970), 630–648 10.1137/1115072 0236.60024 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Nonclassical Estimates of the Rate of Convergence in the Central Limit Theorem Taking into Account Large DeviationsS. Ya. ShorginTheory of Probability & Its Applications, Vol. 27, No. 2 | 17 July 2006AbstractPDF (706 KB)Limit theorems for polylinear formsJournal of Multivariate Analysis, Vol. 9, No. 4 | 1 Dec 1979 Cross Ref A Non-Classical Estimate of the Speed of Convergence in the Multidimensional Central Limit Theorem Taking into Account Large DeviationsS. Ya. ShorginTheory of Probability & Its Applications, Vol. 23, No. 3 | 17 July 2006AbstractPDF (402 KB) Volume 23, Issue 1| 1978Theory of Probability & Its Applications1-226 History Submitted:01 April 1976Published online:17 July 2006 InformationCopyright © 1978 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1123004Article page range:pp. 50-62ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On the Rate of Convergence for the Invariance Principle

On the Rate of Convergence for the Invariance Principle

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

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