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A Stochastic Calculus Proof of the CLT for the L 2 Modulus of Continuity of Local Time

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Book cover Séminaire de Probabilités XLIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2006))

Abstract

We give a stochastic calculus proof of the Central Limit Theorem

$${ \int \nolimits \nolimits {({L}_{t}^{x+h} - {L}_{t}^{x})}^{2}\,dx - 4ht \over {h}^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c{\left (\int \nolimits \nolimits {({L}_{t}^{x})}^{2}\,dx\right )}^{1/2}\,\,\eta $$

as h → 0 for Brownian local time L t x. Here η is an independent normal random variable with mean zero and variance one.

MSC 2000: Primary 60F05, 60J55, 60J65.

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References

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Acknowledgements

This research was supported, in part, by grants from the National Science Foundation and PSC-CUNY.

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Authors and Affiliations

  1. Department of Mathematics, College of Staten Island, CUNY, Staten Island, NY, 10314, USA

    Jay Rosen

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  1. Jay Rosen
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Correspondence to Jay Rosen .

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Editors and Affiliations

  1. Lab. de Probabilités et Modèles Aléatoir, Université Pierre et Marie Curie, place Jussieu 4, Paris Cedex 05, 75252, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Rosen, J. (2011). A Stochastic Calculus Proof of the CLT for the L 2 Modulus of Continuity of Local Time. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIII. Lecture Notes in Mathematics(), vol 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15217-7_3

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