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Teor. Veroyatnost. i Primenen., 1999, Volume 44, Issue 3, Pages 573–588 (Mi tvp804)  

This article is cited in 3 scientific papers (total in 3 papers)

Some bounds on the rate of convergence in the CLT for martingales. II

I. Rinotta, V. I. Rotar'b

a Department of Mathematics, UCSD, CA
b Central Economics and Mathematics Institute, RAS

Abstract: This paper concerns rates of convergence in the central limit theorem (CLT) for the random variables $S_{n}=\sum_{1}^{n}X_{m}$, where $X_{m}$ are martingale-differences. It is known that in the general case one cannot hope for a rate better than $O(n^{-1/8})$ even if the third moments are finite. If the conditional variances satisfy ${\mathbf E}\{X_{m}^2 | X_{1},\ldots, X_{m-1}\}={\mathbf E} X_{m}^2$, the rate in general is no better than $O(n^{-1/4})$, while in the independency case it is of the order $O(n^{-1/2})$. This paper contains a bound which covers simultaneously the cases mentioned as well as some intermediate cases. The bound is presented in terms of some dependency characteristics reflecting the influence of different factors on the rate.

Keywords: central limit theorem, martingales, rate of convergence.

DOI: https://doi.org/10.4213/tvp804

Full text: PDF file (743 kB)

English version:
Theory of Probability and its Applications, 2000, 44:3, 523–536

Bibliographic databases:

Received: 12.08.1997

Citation: I. Rinott, V. I. Rotar', “Some bounds on the rate of convergence in the CLT for martingales. II”, Teor. Veroyatnost. i Primenen., 44:3 (1999), 573–588; Theory Probab. Appl., 44:3 (2000), 523–536

Citation in format AMSBIB
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\by I.~Rinott, V.~I.~Rotar'
\paper Some bounds on the rate of convergence in the CLT for martingales.~II
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\issue 3
\pages 573--588
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\zmath{https://zbmath.org/?q=an:0969.60036}
\transl
\jour Theory Probab. Appl.
\yr 2000
\vol 44
\issue 3
\pages 523--536
\crossref{https://doi.org/10.1137/S0040585X97977744}
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    This publication is cited in the following articles:
    1. El Machkouri M., Ouchti L., “Exact convergence rates in the central limit theorem for a class of martingales”, Bernoulli, 13:4 (2007), 981–999  crossref  mathscinet  zmath  isi  elib  scopus
    2. Roellin A., “Stein's Method in High Dimensions with Applications”, Ann. Inst. Henri Poincare – Probab. Stat., 49:2 (2013), 529–549  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Hafouta Y., Kifer Yu., “Berry?Esseen type estimates for nonconventional sums”, Stoch. Process. Their Appl., 126:8 (2016), 2430–2464  crossref  mathscinet  zmath  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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