Lineability within probability theory settings

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. 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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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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Keywords

  • Lineability
  • Spaceability
  • Probability theory
  • Random variable
  • Stochastic process
  • Martingale

Mathematics Subject Classification

  • 46E10
  • 46E99
  • 60B11