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Manuscript Click here to download Manuscript ultima.tex Click here to view linked References 1 2 3 4 5 6 7 8 9 10 11 LINEABILITY WITHIN PROBABILITY THEORY 12 SETTINGS 13 14 J. ALBERTO CONEJERO, MAR FENOY, MARINA MURILLO-ARCILA, 15 AND JUAN B. SEOANE-SEPÚLVEDA 16 17 18 Abstract. The search of lineability consists on finding large vector 19 spaces of mathematical objects with special properties. Such examples 20 have arisen in the last years in a wide range of settings such as in real 21 and complex analysis, sequence spaces, linear dynamics, norm-attaining 22 functionals, zeros of polynomials in Banach spaces, Dirichlet series, and 23 non-convergent Fourier series, among others. 24 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, 27 and certain stochastic processes. 28 29 30 31 1. Introduction 32 33 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, 35 the search for linear structures of mathematical objects enjoying certain spe36 cial or unexpected properties. Vector spaces and linear algebras are elegant 37 mathematical structures which, at first glance, seem to be “forbidden” to 38 families of “strange” objects. In other words, take a function with some 39 special or (as sometimes it is called) “pathological” property (for exam40 41 ple, the classical nowhere differentiable function, also known as Weierstrass’ 42 monster). Coming up with a concrete example of such a function might 43 be difficult. In fact, it may seem so difficult that if you succeed, you think 44 that there cannot be too many functions of that kind. Probably one can45 not find infinite dimensional vector spaces or infinitely generated algebras 46 of such functions. This is, however, exactly what has been happening in the 47 last years in many fields of mathematics, from Linear Chaos to Real and 48 49 Complex Analysis [3, 5, 12], passing through Set Theory [13] and Linear and 50 Multilinear Algebra, or even Operator Theory [8, 10], Topology, Measure 51 Theory [5], and Abstract Algebra. 52 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 54 exists an infinite dimensional linear space (respectively, infinite dimensional 55 56 2010 Mathematics Subject Classification. 46E10, 46E99, 60B11. 57 Key words and phrases. lineability; spaceability; probability theory; random variable; 58 stochastic process; martingale. 59 1 60 61 62 63 64 65