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The Maximal Variation of a Bounded Martingale and the Central Limit Theorem

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  • DE MEYER, Bernard

    (Laboratoire d'Econométrie, Ecole Polytechnique, Paris and Center for Operations Research and Econometrics (CORE), Université catholique de Louvain (UCL), Louvain la Neuve, Belgium)

Abstract

Mertens and Zamir's (1977) paper is concerned with the asymptotic behaviour of the maximal L[exp.1]-variation [xi1.n(p)] of a [0,1]-valued martingale of length n starting at p. They prove the convergence of [ [xi1.n(p)] / [square root.n]]. to the normal density evaluated at its p-quantile. This paper generalises this result to the conditional L[exp.q]-variation for q [belong] [1,2). The appearance of the normal density remained unexplained in Mertens and Zamir's proof: it appeared there as the solution of a differential equation. Our proof however justifies this normal density as a consequence of a generalisation of the CLT discussed in the second part of this paper.

Suggested Citation

  • DE MEYER, Bernard, 1996. "The Maximal Variation of a Bounded Martingale and the Central Limit Theorem," LIDAM Discussion Papers CORE 1996035, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1996035
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp1996.html
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    Cited by:

    1. Jeffrey Ely & Alexander Frankel & Emir Kamenica, 2015. "Suspense and Surprise," Journal of Political Economy, University of Chicago Press, vol. 123(1), pages 215-260.
    2. Fedor Sandomirskiy, 2014. "Repeated games of incomplete information with large sets of states," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(4), pages 767-789, November.
    3. Fedor Sandomirskiy, 2018. "On Repeated Zero-Sum Games with Incomplete Information and Asymptotically Bounded Values," Dynamic Games and Applications, Springer, vol. 8(1), pages 180-198, March.
    4. De Meyer, Bernard, 2010. "Price dynamics on a stock market with asymmetric information," Games and Economic Behavior, Elsevier, vol. 69(1), pages 42-71, May.

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