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Acta Applicandae Mathematica

, Volume 38, Issue 1, pp 109–129 | Cite as

Large deviations for martingales with some applications

  • A. Račkauskas
Article

Abstract

Let(X i ) be a martingale difference sequence. LetY be a standard normal random variable. We investigate the rate of uniform convergence
$$P\left\{ {\sum\limits_{k = 1}^n {X_k } > \sqrt n r} \right\}/P\{ Y > r\} \to 0asn \to \infty ,$$
asn → ∞, over 0⩽r⩽o(n1/6) in the case of bounded martingale differences. The results are applied to prove large deviations for the ‘baker transformation’. Moderate deviations for martingales are also discussed.

Mathematics subject classifications (1991)

60F10 60G42 Key words Central limit theorem large deviations martingales 

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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • A. Račkauskas
    • 1
  1. 1.Department of MathematicsVilnius UniversityVilniusLithuania

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