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On a nonstandard Brownian motion and its maximal function

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  • Andrade, Bernardo B. de

Abstract

This article uses Radically Elementary Probability Theory (REPT) to prove results about the Wiener walk (the radically elementary Brownian motion) without the technical apparatus required by stochastic integration. The techniques used replace measure-theoretic tools by discrete probability and the rigorous use of infinitesimals. Specifically, REPT is applied to the results in Palacios (The American Statistician, 2008) to calculate certain expectations related to the Wiener walk and its maximal function. Because Palacios uses mostly combinatorics and no measure theory his results carry over through REPT with minimal changes. The paper also presents a construction of the Wiener walk which is intended to mimic the construction of Brownian motion from “continuous” white noise. A brief review of the nonstandard model on which REPT is based is given in the Appendix in order to minimize the need for previous exposure to the subject.

Suggested Citation

  • Andrade, Bernardo B. de, 2015. "On a nonstandard Brownian motion and its maximal function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 429(C), pages 1-9.
    • Handle: RePEc:eee:phsmap:v:429:y:2015:i:c:p:1-9
    DOI: 10.1016/j.physa.2015.01.066
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    References listed on IDEAS

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    1. Palacios, Jose Luis, 2008. "On the Simple Symmetric Random Walk and its Maximal Function," The American Statistician, American Statistical Association, vol. 62, pages 138-140, May.
    2. Robert N. Horn, 1988. "Analysis," Challenge, Taylor & Francis Journals, vol. 31(4), pages 56-58, July.
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