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AN EXTENSION OF THE SUBSE UENCE PRINCIPLE

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 36: Issue 3-4
DOI:
https://doi.org/10.1556/sscmath.36.2000.3-4.6
Author: E. Péter

CHATTERJI, S. D., A principle of subsequences in probability theory: The central limit theorem, Adv. Math. 13 (1974), 31-54; correction, ibid. 14 (1974), 266-269. MR 49 #6312 A principle of subsequences in probability theory

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LIMIT THEOREMS ON THE DIRECT PRODUCT OF A NON-COMPACT LIE GROUP AND A COMPACT GROUP

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 38: Issue 1-4
DOI:
https://doi.org/10.1556/sscmath.38.2001.1-4.21
Authors: G. Pap and Péter Major

2010 PAP, G., Central limit theorems on nilpotent Lie groups, Probab. Math. Statist. 14 (1993), 287-312. MR 96c :22010 Central limit theorems on nilpotent Lie

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Weak convergence of random walks, conditioned to stay away from small sets

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 50: Issue 1
DOI:
https://doi.org/10.1556/sscmath.50.2013.1.1232
Authors: Zsolt Pajor-Gyulai and Domokos Szász

Bolthausen, E. , On a functional central limit theorem for random walks conditioned to stay positive, The Annals of Probability , 4 , No. 3, 480–485, 1976. Bolthausen E. On a functional

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Joint asymptotic behavior of local and occupation times of random walk in higher dimension

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 44: Issue 4
DOI:
https://doi.org/10.1556/sscmath.2007.1035
Authors: Endre Csáki, Antónia Földes, and Pál Révész

Hamana, Y. , On the central limit theorem for the multiple point range of random walk, J. Fac. Sci. Univ. Tokyo 39 (1992), 339–363. MR 93h :60112 Hamana Y. On the central

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A Berry--Esseen theorem for weakly negatively dependent random variables and its applications

Acta Mathematica Hungarica
Volume/Issue:
Volume 110: Issue 4
DOI:
https://doi.org/10.1007/s10474-006-0024-x
Authors: Jianfeng Wang and Lixin Zhang

Summary  

We provide uniform rates of convergence in the central limit theorem for linear negative quadrant dependent (LNQD) random variables. Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\{X_{n},\allowbreak n\ge1\}$ \end{document} be a LNQD sequence of random variables with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $EX_{n}=0$ \end{document}, set \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S_{n}=\sum_{j=1}^{n}X_{j}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $B_{n}^{2}=\text{Var}\, (S_{n})$ \end{document}. We show that \begin{gather*} \sup_{x} \left|P\left(\frac{S_{n}}{B_{n}}<x\right)-\Phi(x)\right|= O\bigg(n^{-\delta/(2+3\delta)}\vee \frac{n^{3\delta^{2}/(4+6\delta)}}{B^{2+\delta}_{n}} \sum_{i=1}^{n} E{|X_{i}|}^{2+\delta}\bigg) \end{gather*} under finite \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(2+\delta)$ \end{document}th moment and a power decay rate of covariances. Moreover, by the truncation method, we obtain a Berry--Esseen type estimate for negatively associated (NA) random variables with only finite second moment. As applications, we obtain another convergence rate result in the central limit theorem and precise asymptotics in the law of the iterated logarithm for NA sequences, and also for LNQD sequences.

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Metric discrepancy results for Erdős-Fortet sequence

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 49: Issue 1
DOI:
https://doi.org/10.1556/sscmath.2011.1186
Authors: Katusi Fukuyama and Sho Miyamoto

Fukuyama, K. , The central limit theorem for Riesz-Raikov sums, Prob. Theory Related Fields , 100 (1994), no. 1, 57–75. MR 1292190 ( 95i :60020) Fukuyama K. The central limit theorem

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Limsup theorems and a uniform LIL for asymptotically negatively associated random sequences

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 49: Issue 2
DOI:
https://doi.org/10.1556/sscmath.49.2012.2.1201
Authors: Yong-Kab Choi and Kyo-Shin Hwang

Zhang, L. X. , A functional central limit theorem for asymptotically negatively dependent random fields, Acta Math. Hungar. , 86 (3) (2000a), 237–259. MR 1756175 ( 2001c :60055) Zhang L. X

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Weak invariance principle for mixing sequences in the domain of attraction of normal law

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 46: Issue 3
DOI:
https://doi.org/10.1556/sscmath.2009.1093
Authors: Raluca Balan and Rafał Kulik

589 Bradley, R. C. , A central limit theorem for stationary ρ -mixing sequences with infinite variance, Ann. Probab. , 16 (1988), 313–332. MR 89a :60053

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Conditions for equivalence between Mallows distance and convergence to stable laws

Acta Mathematica Hungarica
Volume/Issue:
Volume 134: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-011-0101-7
Authors: Chang C. Y. Dorea and Debora B. Ferreira

] Hall , P. , Heyde , C. C. 1980 Martingale Limit Theory and Its Applications Academic Press New York . [6] Johnson , O. , Samworth , R. 2005 Central limit theorem and convergence to

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A general Hsu-Robbins-Erdős Type estimate of tail probabilities of sums of independent identically distributed random variables

Periodica Mathematica Hungarica
Volume/Issue:
Volume 46: Issue 2
DOI:
https://doi.org/10.1023/a:1025992211128
Author: Alexander Pruss

Abstract  

Let X 1,X 2,... be a sequence of independent and identically distributed random variables, and put

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_n = X_1 + \cdot \cdot \cdot + X_n$$ \end{document}
. Under some conditions on the positive sequence τ n and the positive increasing sequence a n, we give necessary and sufficient conditions for the convergence of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\nolimits_{n = 1}^\infty {\tau _n } P\left( {\left| {S_n } \right| \geqslant \varepsilon an} \right)$$ \end{document}
for all & > 0, generalizing Baum and Katz's~(1965) generalization of the Hsu–Robbins–Erds (1947, 1949) law of large numbers, also allowing us to characterize the convergence of the above series in the case where τn = n -1and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$an = \left( {n\log n} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ \end{document}
for n ≤ 2, thereby answering a question of Spătaru. Moreover, some results for non-identically distributed independent random variables are obtained by a recent comparison inequality. Our basic method is to use a central limit theorem estimate of Nagaev (1965) combined with the Hoffman-Jrgensen inequality~(1974).

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