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Teor. Veroyatnost. i Primenen., 1980, Volume 25, Issue 4, Pages 800–818 (Mi tvp1233)  

This article is cited in 18 scientific papers (total in 18 papers)

On the rate of convergence in the central limit theorem for weakly dependent random variables

A. N. Tihomirov

Syktyvkar

Abstract: Let $X_1,X_2,…$ be a stationary sequence of random variables with $\mathbf EX_1=0$, $\mathbf E|X_1|^3<\infty$. Let
\begin{gather*} \sigma^2_n=\mathbf E(\sum_{j=1}^n X_j)^2,\qquad F_n(x)=\mathbf P\{\sigma_n^{-1}\sum_{j=1}^n X_j<x\},
\Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^x e^{-y^2/2} dy,\qquad \Delta_n=\sup|F_n(x)-\Phi(x)|. \end{gather*}
We prove that if the sequence $X_n$ satisfies a strong mixing condition and if its mixing coefficient decreases exponentially then
$$ \Delta_n=O(n^{-1/2}\ln^2n). $$
For the case of $m$-dependent variables we prove that
$$ \Delta_n=O(m^2n^{-1/2}). $$


Full text: PDF file (923 kB)

English version:
Theory of Probability and its Applications, 1981, 25:4, 790–809

Bibliographic databases:

Received: 18.01.1979

Citation: A. N. Tihomirov, “On the rate of convergence in the central limit theorem for weakly dependent random variables”, Teor. Veroyatnost. i Primenen., 25:4 (1980), 800–818; Theory Probab. Appl., 25:4 (1981), 790–809

Citation in format AMSBIB
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\by A.~N.~Tihomirov
\paper On the rate of convergence in the central limit theorem for weakly dependent random variables
\jour Teor. Veroyatnost. i Primenen.
\yr 1980
\vol 25
\issue 4
\pages 800--818
\mathnet{http://mi.mathnet.ru/tvp1233}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=595140}
\zmath{https://zbmath.org/?q=an:0471.60030|0448.60019}
\transl
\jour Theory Probab. Appl.
\yr 1981
\vol 25
\issue 4
\pages 790--809
\crossref{https://doi.org/10.1137/1125092}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1980MK50200010}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Sunklodas J., “On a lower bound of the uniform distance in the central limit theorem for phi–mixing random variables”, Acta Applicandae Mathematicae, 58:1–3 (1999), 327–341  crossref  mathscinet  zmath  isi
    2. Sh. K. Formanov, “The Stein–Tikhomirov Method and a Nonclassical Central Limit Theorem”, Math. Notes, 71:4 (2002), 550–555  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Chen L.H.Y., Shao Q.M., “Normal approximation under local dependence”, Annals of Probability, 32:3A (2004), 1985–2028  crossref  mathscinet  zmath  isi
    4. T. P. Kazanchyan, “Otsenka skorostn skhodimosti v predelnoi teoreme Erdesha–Katsa dlya zavisimykh sluchainykh velichin”, Uch. zapiski EGU, ser. Fizika i Matematika, 2004, no. 2, 20–26  mathnet
    5. Sh. K. Formanov, “On the Stein–Tikhomirov method and its applications in nonclassical limit theorems”, Discrete Math. Appl., 17:1 (2007), 23–36  mathnet  crossref  crossref  mathscinet  zmath  elib
    6. J. Sunklodas, “On normal approximation for strongly mixing random fields”, Theory Probab. Appl., 52:1 (2008), 125–132  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Bentkus V., Sunklodas J.K., “On normal approximations to strongly mixing random fields”, Publicationes Mathematicae–Debrecen, 70:3–4 (2007), 253–270  mathscinet  zmath  isi
    8. Hormann S., “Berry-Esseen bounds for econometric time series”, Alea-Latin American Journal of Probability and Mathematical Statistics, 6 (2009), 377–397  isi
    9. Petrauskiene J., Cekanavicius V., “Compound Poisson Approximations for Sums of 1-Dependent Random Variables. I”, Lith Math J, 50:3 (2010), 323–336  crossref  mathscinet  zmath  isi
    10. Sunklodas J.K., “SOME ESTIMATES OF THE NORMAL APPROXIMATION FOR phi-MIXING RANDOM VARIABLES”, Lith Math J, 51:2 (2011), 260–273  crossref  isi
    11. Grin A.G., “On the accuracy of normal approximation of distributions of sums of dependent random variables”, Matematicheskie struktury i modelirovanie, 2012, no. 26, 5–19  elib
    12. Cekanavicius V., “Approximation Methods in Probability Theory”, Approximation Methods in Probability Theory, Universitext, Springer International Publishing Ag, 2016, 1–274  crossref  isi
    13. Goetze F. Naumov A. Tikhomirov A., “Distribution of Linear Statistics of Singular Values of the Product of Random Matrices”, Bernoulli, 23:4B (2017), 3067–3113  crossref  isi
    14. Mangia M. Pareschi F. Rovatti R. Setti G., “Low-Cost Security of Iot Sensor Nodes With Rakeness-Based Compressed Sensing: Statistical and Known-Plaintext Attacks”, IEEE Trans. Inf. Forensic Secur., 13:2 (2018), 327–340  crossref  isi
    15. Goldstein L. Wiroonsri N., “Stein'S Method For Positively Associated Random Variables With Applications to the Ising and Voter Models, Bond Percolation, and Contact Process”, Ann. Inst. Henri Poincare-Probab. Stat., 54:1 (2018), 385–421  crossref  isi
    16. Theory Probab. Appl., 63:3 (2019), 479–499  mathnet  crossref  crossref  isi  elib
    17. Ibragimov I.A. Lifshits M.A. Nazarov A.I. Zaporozhets D.N., “On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables”, Vestn. St Petersb. Univ.-Math., 51:3 (2018), 213–236  crossref  isi  scopus
    18. Raic M., “A Multivariate Berry-Esseen Theorem With Explicit Constants”, Bernoulli, 25:4A (2019), 2824–2853  crossref  isi
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