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LINEABILITY WITHIN PROBABILITY THEORY
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SETTINGS
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J. ALBERTO CONEJERO, MAR FENOY, MARINA MURILLO-ARCILA,
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AND JUAN B. SEOANE-SEPÚLVEDA
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Abstract. The search of lineability consists on finding large vector
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spaces of mathematical objects with special properties. Such examples
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have arisen in the last years in a wide range of settings such as in real
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and complex analysis, sequence spaces, linear dynamics, norm-attaining
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functionals, zeros of polynomials in Banach spaces, Dirichlet series, and
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non-convergent Fourier series, among others.
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In this paper we present the novelty of linking this notion of lineabil25
ity to the area of Probability Theory by providing positive (and neg26
ative) results within the framework of martingales, random variables,
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and certain stochastic processes.
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1. Introduction
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Since the beginning of the 21st century many authors have become inter34
ested in the study of linearity within non linear settings or, in other words,
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the search for linear structures of mathematical objects enjoying certain spe36
cial or unexpected properties. Vector spaces and linear algebras are elegant
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mathematical structures which, at first glance, seem to be “forbidden” to
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families of “strange” objects. In other words, take a function with some
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special or (as sometimes it is called) “pathological” property (for exam40
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ple, the classical nowhere differentiable function, also known as Weierstrass’
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monster). Coming up with a concrete example of such a function might
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be difficult. In fact, it may seem so difficult that if you succeed, you think
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that there cannot be too many functions of that kind. Probably one can45
not find infinite dimensional vector spaces or infinitely generated algebras
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of such functions. This is, however, exactly what has been happening in the
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last years in many fields of mathematics, from Linear Chaos to Real and
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Complex Analysis [3, 5, 12], passing through Set Theory [13] and Linear and
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Multilinear Algebra, or even Operator Theory [8, 10], Topology, Measure
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Theory [5], and Abstract Algebra.
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Recall that, as it nowadays is common terminology, a subset M of a topo53
logical vector space X is called lineable (respectively, spaceable) in X if there
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exists an infinite dimensional linear space (respectively, infinite dimensional
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2010 Mathematics Subject Classification. 46E10, 46E99, 60B11.
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Key
words and phrases. lineability; spaceability; probability theory; random variable;
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stochastic
process; martingale.
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