Lineability within probability theory settings

[EN] 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.This work was partially supported by Ministerio de Educacion, 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 Economia y Competitividad: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.Conejero, JA.; Fenoy, M.; Murillo Arcila, M.; Seoane Sepúlveda, JB. (2017). Lineability within probability theory settings. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 111(3):673-684. https://doi.org/10.1007/s13398-016-0318-yS6736841113Aizpuru, A., Pérez-Eslava, C., García-Pacheco, F.J., Seoane-Sepúlveda, J.B.: Lineability and coneability of discontinuous functions on $$\mathbb{R}$$ R . Publ. Math. Debrecen 72(1–2), 129–139 (2008)Aron, R., Gurariy, V.I., Seoane, J.B.: Lineability and spaceability of sets of functions on $$\mathbb{R}$$ R . Proc. Am. Math. Soc. 133(3), 795–803 (2005, electronic)Aron, R.M., González, L.B., Pellegrino, D.M., Sepúlveda J.B.S.: Lineability: the search for linearity in mathematics. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2016)Ash, R.B.: Real analysis and probability. Probability and mathematical statistics, No. 11. Academic Press, New York-London (1972)Barbieri, G., García-Pacheco, F.J., Puglisi, D.: Lineability and spaceability on vector-measure spaces. Stud. Math. 219(2), 155–161 (2013)Bernal-González, L., Cabrera, M.O.: Lineability criteria, with applications. J. Funct. Anal. 266(6), 3997–4025 (2014)Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. (N.S.), 51(1), 71–130 (2014)Berndt, B.C.: What is a $$q$$ q -series? In: Ramanujan rediscovered, Ramanujan Math. Soc. Lect. Notes Ser., vol. 14, pp. 31–51. Ramanujan Math. Soc., Mysore (2010)Bertoloto, F.J., Botelho, G., Fávaro, V.V., Jatobá, A.M.: Hypercyclicity of convolution operators on spaces of entire functions. Ann. Inst. Fourier (Grenoble) 63(4), 1263–1283 (2013)Billingsley, P.: Probability and measure. Wiley Series in Probability and Mathematical Statistics, 3rd edn, A Wiley-Interscience Publication. Wiley, New York (1995)Botelho, G., Fávaro, V.V.: Constructing Banach spaces of vector-valued sequences with special properties. Mich. Math. J. 64(3), 539–554 (2015)Cariello, D., Seoane-Sepúlveda, J.B.: Basic sequences and spaceability in $$\ell _p$$ ℓ p spaces. J. Funct. Anal. 266(6), 3797–3814 (2014)Drewnowski, L., Lipecki, Z.: On vector measures which have everywhere infinite variation or noncompact range. Dissertationes Math. (Rozprawy Mat.) 339, 39 (1995)Dugundji, J.: Topology. Allyn and Bacon, Inc., Boston, Mass.-London-Sydney (1978, Reprinting of the 1966 original, Allyn and Bacon Series in Advanced Mathematics)Enflo, P.H., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Some results and open questions on spaceability in function spaces. Trans. Am. Math. Soc. 366(2), 611–625 (2014)Fonf, V.P., Zanco, C.: Almost overcomplete and almost overtotal sequences in Banach spaces. J. Math. Anal. Appl. 420(1), 94–101 (2014)Gámez-Merino, J.L., Seoane-Sepúlveda, J.B.: An undecidable case of lineability in $$\mathbb{R}^{\mathbb{R}}$$ R R . J. Math. Anal. Appl. 401(2), 959–962 (2013)Gurariĭ, V.I.: Linear spaces composed of everywhere nondifferentiable functions. C. R. Acad. Bulgare Sci. 44(5), 13–16 (1991)Muñoz-Fernández, G.A., Palmberg, N., Puglisi, D., Seoane-Sepúlveda, J.B.: Lineability in subsets of measure and function spaces. Linear Algebra Appl. 428(11–12), 2805–2812 (2008)Walsh, J.B.: Martingales with a multidimensional parameter and stochastic integrals in the plane. In: Lectures in probability and statistics (Santiago de Chile, 1986), Lecture Notes in Math., vol. 1215, pp. 329–491. Springer, Berlin (1986)Wise, G.L., Hall, E.B.: Counterexamples in probability and real analysis. The Clarendon Press, Oxford University Press, New York (1993