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Central limit theorems for logarithmic averages

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 38: Issue 1-4
DOI:
https://doi.org/10.1556/sscmath.38.2001.1-4.6
Authors: I. Berkes, L. Horváth, and X. Chen

everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 561-574. MR 89i :60045 An almost everywhere central limit theorem Math. Proc. Cambridge Philos. Soc

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A moment approach for the almost sure central limit theorem for martingales

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 45: Issue 1
DOI:
https://doi.org/10.1556/sscmath.2007.1047
Authors: Bernard Bercu and Jean-Claude Fort

134 Bercu, B. , On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications, Stochastic Process. Appl. 111 (2004

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A central limit theorem for normalized products of random matrices

Periodica Mathematica Hungarica
Volume/Issue:
Volume 56: Issue 2
DOI:
https://doi.org/10.1007/s10998-008-6183-3
Authors: Rolando Cavazos-Cadena and Daniel Hernández-Hernández

Abstract  

This note concerns the asymptotic behavior of a Markov process obtained from normalized products of independent and identically distributed random matrices. The weak convergence of this process is proved, as well as the law of large numbers and the central limit theorem.

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Almost sure central limit theorems for weighted dependent sequences

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 2
DOI:
https://doi.org/10.1556/012.2019.56.2.1426
Author: Khurelbaatar Gonchigdanzan

] B rosamler , G. A. , An almost everywhere central limit theorem , Math. Proc. Camb. Phil. Soc. , 104 ( 1988 ), 561 – 574 . [4] G onchigdanzan , K

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Martingale random central limit theorems

Acta Mathematica Hungarica
Volume/Issue:
Volume 34: Issue 1-2
DOI:
https://doi.org/10.1007/bf01902602
Author: Z. Rychlik
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Random central limit theorems for martingales

Acta Mathematica Hungarica
Volume/Issue:
Volume 20: Issue 1-2
DOI:
https://doi.org/10.1007/bf01894583
Author: B. L. S. Prakasa Rao
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A converse to a central limit theorem of B. Gyires

Acta Mathematica Hungarica
Volume/Issue:
Volume 31: Issue 1-2
DOI:
https://doi.org/10.1007/bf01896074
Author: D. B. Wolfson
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On the universal A.S. central limit theorem

Acta Mathematica Hungarica
Volume/Issue:
Volume 116: Issue 4
DOI:
https://doi.org/10.1007/s10474-007-6070-1
Author: S. Hörmann

Abstract  

Let (X k) be a sequence of independent r.v.’s such that for some measurable functions gk : R k → R a weak limit theorem of the form

\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} $$g_k (X_1 , \ldots ,X_k )\xrightarrow{\mathcal{L}}G$$ \end{document}
holds with some distribution function G. By a general result of Berkes and Csáki (“universal ASCLT”), under mild technical conditions the strong analogue
\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} $$\frac{1} {{D_N }}\sum\limits_{k = 1}^N {d_k I\left\{ {g_k (X_1 , \ldots ,X_k ) \leqq x} \right\} \to G(x)} a.s.$$ \end{document}
 is also valid, where (d k) is a logarithmic weight sequence and D N = ∑k=1 N d k. In this paper we extend the last result for a very large class of weight sequences (d k), leading to considerably sharper results. We show that logarithmic weights, used traditionally in a.s. central limit theory, are far from optimal and the theory remains valid with averaging procedures much closer to, in some cases even identical with, ordinary averages.

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On central limit theorems for martingale triangular arrays

Acta Mathematica Hungarica
Volume/Issue:
Volume 31: Issue 3-4
DOI:
https://doi.org/10.1007/bf01901971
Authors: P. Gaenssler, J. Strobel, and W. Stute
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The central limit theorem for stationary associated sequences

Acta Mathematica Hungarica
Volume/Issue:
Volume 97: Issue 1-2
DOI:
https://doi.org/10.1023/a:1020802711047
Authors: S. Louhichi and Ph. Soulier
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