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Lineability within probability theory settings

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The search of lineability consists on finding large vector spaces of mathematical objects with special properties. Such examples have arisen in the last years in a wide range of settings such as in real and complex analysis, sequence spaces, linear dynamics, norm-attaining functionals, zeros of polynomials in Banach spaces, Dirichlet series, and non-convergent Fourier series, among others. In this paper we present the novelty of linking this notion of lineability to the area of Probability Theory by providing positive (and negative) results within the framework of martingales, random variables, and certain stochastic processes.

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Notes

  1. By \(\mathcal {F}_n=\sigma (X_1,\ldots ,X_n)\) we mean the smallest \(\sigma \)-algebra in which \(\{X_i : i \le n\}\) are measurable.

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Acknowledgments

This work was partially supported by Ministerio de Educación, Cultura y Deporte, projects MTM2013-47093-P and MTM2015-65825-P, by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministerio de Economía y Competitividad: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

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Authors and Affiliations

  1. Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Universitat Politècnica de València, 46022, Valencia, Spain

    J. Alberto Conejero

  2. Departamento de Estadística e Investigación Operativa, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Mar Fenoy

  3. BCAM-Basque Center for Applied Mathematics, Mazarredo, 14, 48009, Bilbao, Basque Country, Spain

    Marina Murillo-Arcila

  4. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera 13–15, Campus de Cantoblanco, UAM, 28049, Madrid, Spain

    Juan B. Seoane-Sepúlveda

  5. Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Juan B. Seoane-Sepúlveda

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  1. J. Alberto Conejero
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  2. Mar Fenoy
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  3. Marina Murillo-Arcila
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  4. Juan B. Seoane-Sepúlveda
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Correspondence to J. Alberto Conejero.

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Dedicated to Prof. Manuel Maestre on the occasion of his 60th birthday.

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Conejero, J.A., Fenoy, M., Murillo-Arcila, M. et al. Lineability within probability theory settings. RACSAM 111, 673–684 (2017). https://doi.org/10.1007/s13398-016-0318-y

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  • DOI: https://doi.org/10.1007/s13398-016-0318-y

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