Convergence In Central Limit Theorem Research Articles

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

Previous article Next article Uniform Estimates of the Rate of Convergence in the Multi-Dimensional Central Limit TheoremV. V. SenatovV. V. Senatovhttps://doi.org/10.1137/1125089PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. N. Vakhaniya, Probability Distributions in Linear Spaces, Metsniereba, Tbilisi, 1971, (In Russian.) Google Scholar[2] V. M. Zolotarev, Approximation of distributions of sums of independent random variables with values in infinite-dimensional spaces, Theory Prob. Appl., 21 (1976), 721–737 LinkGoogle Scholar[3] V. M. Zolotarev, Metric distances in spaces of random variables and of their distributions, Mat. Sb. (N.S.), 101(143) (1976), 416–454, 456, (In Russian.) 57:7720 Google Scholar[4] V. M. Zolotarev, Ideal metrics in the problem of approximating distributions of sums of independent random variables, Theory Prob. Appl., 22 (1977), 433–449 LinkGoogle Scholar[5] V. M. Zolotarev, Properties and relations of certain types of metrics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 87 (1979), 18–35, 206, 212, (In Russian.) 81a:60008 0417.60007 Google Scholar[6] S. V. Nagaev, An estimate of the remainder term in the multidimensional central limit theorem, Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975), Springer, Berlin, 1976, 419–438. Lecture Notes in Math., Vol. 550 56:1416 0363.60024 Google Scholar[7] V. V. Sazonov, On a bound for the rate of convergence in the multidimensional central limit theorem, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, Univ. California Press, Berkeley, Calif., 1972, 563–581 53:4185 0274.60019 Google Scholar[8] V. V. Sazonov and , V. G. Mirantsev, On the central limit theorem in $R^{k}$, Theory Prob. Appl., 23 (1978), 407–411 LinkGoogle Scholar[9] V. V. Ul'yanov, On a more precise convergence rate of estimates in the central limit theorem, Theory Prob. Appl., 23 (1978), 660–663 LinkGoogle Scholar[10] V. V. Ul'yanov, Letter to the editors: “Some improvements of estimates for the rate of convergence in the central limit theorem”, Teor. Veroyatnost. i Primenen., 24 (1979), 236– 80k:60033b Google Scholar[11] V. V. Sazonov and , V. V. Ul'yanov, On the speed of convergence in the central limit theorem, Adv. Appl. Probab., 11 (1979), 269–270 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Central limit theorem and bootstrap approximation in high dimensions: Near 1/n rates via implicit smoothingThe Annals of Statistics, Vol. 50, No. 5 Cross Ref Berry–Esseen bounds for multivariate nonlinear statistics with applications to M-estimators and stochastic gradient descent algorithmsBernoulli, Vol. 28, No. 3 Cross Ref The CLT in high dimensions: Quantitative bounds via martingale embeddingThe Annals of Probability, Vol. 48, No. 5 Cross Ref A high-dimensional CLT in $$\mathcal {W}_2$$ W 2 distance with near optimal convergence rate24 March 2017 | Probability Theory and Related Fields, Vol. 170, No. 3-4 Cross Ref Ideal Metrics with Respect to Summation Scheme for i.i.d. Random Variables12 November 2012 Cross Ref Ideal Metrics and Rate of Convergence in the CLT for Random Motions12 November 2012 Cross Ref A Structural Classification of Probability Distances12 November 2012 Cross Ref Asymptotically optimal Berry–Esseen-type bounds for distributions with an absolutely continuous partJournal of Statistical Planning and Inference, Vol. 141, No. 3 Cross Ref Comments on: Goodness-of-fit tests in mixed models14 May 2009 | TEST, Vol. 18, No. 2 Cross Ref Sums of Independent Random Vectors: Proximity Estimating22 November 2006 | Stochastic Models, Vol. 22, No. 4 Cross Ref A general limit theorem for recursive algorithms and combinatorial structuresThe Annals of Applied Probability, Vol. 14, No. 1 Cross Ref A note on the multivariate local limit theoremStatistics & Probability Letters, Vol. 59, No. 3 Cross Ref On the distance between convex-ordered random variables, with applications1 July 2016 | Advances in Applied Probability, Vol. 34, No. 2 Cross Ref Goodness-of-fit tests for mixed model diagnosticsThe Annals of Statistics, Vol. 29, No. 4 Cross Ref Estimates of stability of geometric convolutionsApplied Mathematics Letters, Vol. 12, No. 5 Cross Ref Edgeworth expansions in very-high-dimensional problemsJournal of Statistical Planning and Inference, Vol. 70, No. 1 Cross Ref Bibliography Cross Ref Rates of convergence in the operator-stable limit theoremJournal of Theoretical Probability, Vol. 9, No. 1 Cross Ref An extension of the kantorovich-rubinstein mass-transshipment problemNumerical Functional Analysis and Optimization, Vol. 16, No. 5-6 Cross Ref On the Rate of Convergence in the CLT with Respect to the Kantorovich Metric Cross Ref Rate of Convergence in the Central Limit Theorem for Generalized Convolutions Cross Ref A new ideal metric with applications to multivariate stable limit theoremsProbability Theory and Related Fields, Vol. 94, No. 2 Cross Ref Approximation of sums by compound Poisson distributions with respect to stop-loss distances1 July 2016 | Advances in Applied Probability, Vol. 22, No. 2 Cross Ref New duality theorems for Marginal problems with some applications in stochastics29 September 2006 Cross Ref A central limit theorem applicable to robust regression estimatorsJournal of Multivariate Analysis, Vol. 22, No. 1 Cross Ref Four Examples of Lower Estimates in the Multi-Dimensional Central Limit TheoremV. V. Senatov28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 4AbstractPDF (828 KB)On the central limit theorem in R p when p→∞Probability Theory and Related Fields, Vol. 73, No. 4 Cross Ref VIII Seminar on Problems of Stability of Stochastic Models17 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 3AbstractPDF (1130 KB)On the Dependency of Estimates of the Convergence Rate in the Central Limit Theorem on the Covariance Operator of the SummandsV. V. Senatov17 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 2AbstractPDF (383 KB)Estimates of convergence rate in Renyi's theoremJournal of Soviet Mathematics, Vol. 32, No. 6 Cross Ref An Estimate of the Lévy–Prokhorov MetricV. V. Senatov17 July 2006 | Theory of Probability & Its Applications, Vol. 29, No. 1AbstractPDF (559 KB)On Estimating the Rate of Convergence in the Central Limit Theorem Over a System of Balls in $R^k $V. V. Senatov17 July 2006 | Theory of Probability & Its Applications, Vol. 28, No. 2AbstractPDF (361 KB) Volume 25, Issue 4| 1981Theory of Probability & Its Applications History Submitted:03 April 1980Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1125089Article page range:pp. 745-759ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

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

Next article On the Rate of Convergence for the Invariance PrincipleA. A. BorovkovA. A. Borovkovhttps://doi.org/10.1137/1118025PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. I. Gikhman and , A. V. Skorokhod, Introduction to the theory of random processes, Translated from the Russian by Scripta Technica, Inc, W. B. Saunders Co., Philadelphia, Pa., 1969xiii+516 MR0247660 Google Scholar[2] Yu. V. Prohorov, Convergence of random processes and limit theorems in probability theory, Theory Prob. Applications, 1 (1956), 157–214 10.1137/1101016 LinkGoogle Scholar[3] Walter A. Rosenkrantz, On rates of convergence for the invariance principle, Trans. Amer. Math. Soc., 129 (1967), 542–552 MR0215347 0201.50902 CrossrefGoogle Scholar[4] C. C. Heyde, On extended rate of convergence results for the invariance principle, Ann. Math. Statist., 40 (1969), 2178–2179 0193.45103 CrossrefGoogle Scholar[5] A. V. Skorokhod, Studies in the theory of random processes, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965viii+199 MR0185620 0146.37701 Google Scholar[6] S. V. Nagaev, On the speed of convergence in a boundary value problem, I, II, Theory Prob. Applications, 15 (1970), 163–186, 403–729 Google Scholar[7] A. V. Skorokhod, A limit theorem for sums of independent random variables, DAN SSSR, 133 (1960), 34–35, (In Russian.) 0104.37003 Google Scholar[8] S. S. Ebralidze, Inequalities for the probabilities of large deviations in the multidimensional case, Theory Prob. Applications, 16 (1971), 733–741 10.1137/1116087 0251.60017 LinkGoogle Scholar[9] S. V. Nagaev and , A. Kh. Fuk, Probability inequalities for sums of independent random variables, Theory Prob. Applications, 16 (1971), 643–660 10.1137/1116071 0259.60024 LinkGoogle Scholar[10] A. A. Borovkov, Notes on inequalities for sums of independent variables, Theory Prob. Applications, 17 (1972), 556–557 10.1137/1117068 0283.60054 LinkGoogle Scholar[11] A. A. Borovkov, The convergence of distributions of functionals of stochastic processes, Uspehi Mat. Nauk, 27 (1972), 3–41, (In Russian.) MR0400325 Google Scholar[12] S. V. Nagaev, On the speed of convergence of the distribution of maximum sums of independent variables, Theory Prob. Applications, 15 (1970), 309–314 10.1137/1115036 0227.60017 LinkGoogle Scholar[13] T. V. Arak, Masters Thesis, On certain question of statistically random processes and on the rate of convergence in certain boundary value problems, Dissertation, Leningrad, 1972, (In Russian.) Google Scholar Next article FiguresRelatedReferencesCited byDetails Prokhorov Distance with Rates of Convergence under Sublinear ExpectationsQ. Zhou, A. I. Sakhanenko, and J. Guo5 February 2021 | Theory of Probability & Its Applications, Vol. 65, No. 4AbstractPDF (565 KB)Prokhorov distance with rates of convergence under sublinear expectations22 October 2020 | Теория вероятностей и ее применения, Vol. 65, No. 4 Cross Ref An invariance principle for the stochastic heat equation7 May 2018 | Stochastics and Partial Differential Equations: Analysis and Computations, Vol. 6, No. 4 Cross Ref The extended large deviation principle for a process with independent increments24 June 2017 | Siberian Mathematical Journal, Vol. 58, No. 3 Cross Ref On the Rate of Convergence of Loop-Erased Random Walk to SLE28 February 2013 | Communications in Mathematical Physics, Vol. 318, No. 2 Cross Ref Точность сильной гауссовской аппроксимации для сумм независимых случайных векторовУспехи математических наук, Vol. 68, No. 4(412) Cross Ref Estimates for the rate of strong approximation in Hilbert space24 August 2011 | Siberian Mathematical Journal, Vol. 52, No. 4 Cross Ref A general estimate in the invariance principle24 August 2011 | Siberian Mathematical Journal, Vol. 52, No. 4 Cross Ref Bounds for the Rate of Strong Approximation in the Multidimensional Invariance PrincipleF. Götze and A. Yu. Zaitsev27 February 2009 | Theory of Probability & Its Applications, Vol. 53, No. 1AbstractPDF (249 KB)Estimates in the invariance principle in terms of truncated power momentsSiberian Mathematical Journal, Vol. 47, No. 6 Cross Ref Applications of permutations to the simulations of critical valuesJournal of Nonparametric Statistics, Vol. 16, No. 1-2 Cross Ref Rates of convergence for U-statistic processes and their bootstrapped versionsJournal of Statistical Planning and Inference, Vol. 102, No. 2 Cross Ref Multidimensional Version of a Result of Sakhanenko in the Invariance Principle for Vectors with Finite Exponential Moments. IIA. Yu. Zaitsev25 July 2006 | Theory of Probability & Its Applications, Vol. 46, No. 3AbstractPDF (254 KB)Multidimensional Version of a Result of Sakhanenko in the Invariance Principle for Vectors with Finite Exponential Moments. IIIA. Yu. Zaitsev25 July 2006 | Theory of Probability & Its Applications, Vol. 46, No. 4AbstractPDF (230 KB)Multidimensional Version of a Result of Sakhanenko in the Invariance Principle for Vectors with Finite Exponential Moments. IA. Yu. Zaitsev25 July 2006 | Theory of Probability & Its Applications, Vol. 45, No. 4AbstractPDF (218 KB)Exponential and polynomial tailbounds for change-point estimatorsJournal of Statistical Planning and Inference, Vol. 92, No. 1-2 Cross Ref Short Communication:Optimal Bounds for the Prokhorov Distance of the Miller--Sen Process and Brownian MotionD. Ferger17 February 2012 | Theory of Probability & Its Applications, Vol. 42, No. 1AbstractPDF (107 KB)Strong approximation of semimartingales and statistical processesProbability Theory and Related Fields, Vol. 104, No. 4 Cross Ref A Refinement of Asymptotics in the Prokhorov–Donsker Invariance Principle for Integral FunctionalsN. K. 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Borisov17 July 2006 | Theory of Probability & Its Applications, Vol. 21, No. 2AbstractPDF (1074 KB) Volume 18, Issue 2| 1974Theory of Probability & Its Applications History Submitted:10 May 1972Published online:17 July 2006 InformationCopyright © 1974 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1118025Article page range:pp. 207-225ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

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

Previous article Next article A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central TheoremV. I. Rotar’V. I. Rotar’https://doi.org/10.1137/1115072PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Harald Bergström, On the central limit theorem in the space $R\sb k, k>1$, Skand. Aktuarietidskr., 28 (1945), 106–127 MR0015704 Google Scholar[2] V. V. Sazonov, On the multi-dimensional central limit theorem, Sankhyā Ser. A, 30 (1968), 181–204 MR0236979 0209.49601 Google Scholar[3] V. V. Sazanov, Masters Thesis, Studies on multi-dimensional and infinite-dimensional limit theorems of probability theory, Doctoral dissertation, Moscow, 1968, (In Russian.) Google Scholar[4] V. I. Paulauskas, Masters Thesis, Estimation of the remainder in the multi-dimensional central limit theorem, Candidate dissertation, Vil'nyus, 1968, (In Russian.) Google Scholar[5] Carl-Gustav Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math., 77 (1945), 1–125 MR0014626 0060.28705 CrossrefGoogle Scholar[6] R. Ranga Rao, On the central limit theorem in $R\sb{k}$, Bull. Amer. Math. Soc., 67 (1961), 359–361 MR0133150 CrossrefGoogle Scholar[7] S. M. Sadikova, Two-dimensional analogues of Esseen's inequality with applications to the central limit theorem, Theory Prob. Applications, 11 (1966), 325–335 10.1137/1111035 LinkGoogle Scholar[8] S. M. Sadikova, A distance between distributions, connected with their values on convex sets, Dokl. Akad. Nauk SSSR, 176 (1967), 787–789, (In Russian.) MR0220331 Google Scholar[9] S. M. Sadikova, On the multi-dimensional central limit theorem, Theory Prob. Applications, 13 (1968), 164–170 10.1137/1113015 0196.20503 LinkGoogle Scholar[10] Bengt von Bahr, On the central limit theorem in $R\sb{k}$, Ark. Mat., 7 (1967), 61–69 (1967) MR0224128 0221.60013 Google Scholar[11] Bengt von Bahr, Multi-dimensional integral limit theorems, Ark. Mat., 7 (1967), 71–88 (1967) MR0228044 0221.60014 Google Scholar[12] R. N. Bhattacharya, Rates of weak convergence for the multi-dimensional central limit theorem, Theory Prob. Applications, 15 (1970), 68–86 10.1137/1115005 0216.46302 LinkGoogle Scholar[13] L. D. Meshalkin and , B. A. Rogozin, An estimate for the distance between distribution functions on the basis of the closeness of their characteristic functions and its application to the central limit theorem, Limit Theorems of Probability Theory, Tashkent, 1963, 49–55, (In Russian.) Google Scholar[14] S. V. Nagaev, Some limit theorems for large deviations, Theory Prob. Applications, 10 (1965), 214–235 10.1137/1110027 0144.18704 LinkGoogle Scholar[15] A. Bikyalis, Estimates for the remainder in the central limit theorem, Litovskii Mat. Sb., 6 (1966), 321–346, (In Russian.) 0139.35302 Google Scholar[16] L. V. Osipov and , V. V. Petrov, An estimate of the remainder term in the central limit theorem, Theory Prob. Applications, 12 (1967), 281–286 10.1137/1112030 0185.46801 LinkGoogle Scholar[17] L. V. Osipov, Asymptotic expansions in the central limit theorem, Vestnik Leningrad. Univ., 22 (1967), 45–62, (In Russian.) MR0221568 0189.18003 Google Scholar[18] V. I. Rotar', On the speed of convergence in the multi-dimensional central limit theorem, Theory Prob. Applications, 15 (1970), 354–356 10.1137/1115044 0227.60018 LinkGoogle Scholar[19] S. Bochner, Vorlesungen über Fouriersche Integrale, Akademische Verlagsgesellschaft, Leipzig, 1932 0006.11001 Google Scholar[20] A. Bikjalis, Asymptotic expansions for the densities and distributions of sums of independent identically distributed random vectors, Litovsk. Mat. Sb., 8 (1968), 405–422, (In Russian.) MR0245069 Google Scholar[21] Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963xvi+685 MR0203748 0108.14202 Google Scholar[22] A. Bikjalis, Remainder terms in asymptotic expansions for characteristic functions and their derivatives, Litovsk. Mat. Sb., 7 (1967), 571–582 (1968), (In Russian.) MR0236970 Google Scholar[23] A. Bikyalis, On multi-dimensional characteristic functions, Litovsk. Mat. Sb., 8 (1968), 21–39, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Compact Law of the Iterated Logarithm for Matrix-Normalized Sums of Random VectorsTheory of Probability & Its Applications, Vol. 52, No. 4 | 19 November 2008AbstractPDF (197 KB)Rate of convergence in the multidimensional central limit theorem for stationary processes. 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