Random Variables: Topology and Geometry

Pablo Koch-Medina and Cosimo Munari
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Pablo Koch-Medina: University of Zurich, Department of Banking and Finance
Cosimo Munari: University of Zurich, Department of Banking and Finance

Chapter 3 in Market-Consistent Prices, 2020, pp 59-82 from Springer

Abstract: Abstract We have already equipped the set of random variables defined on a given sample space with the structure of an ordered vector space. Once a probability measure is specified, one can define a family of norms, the so-called p-norms, on the space of random variables. One of these norms, namely the 2-norm, arises from an inner product. This additional structure allows us to introduce a variety of powerful topological notions such as convergence and continuity, as well as geometrical notions such as orthogonality. To carry out this program we need to assume that the underlying probability space does not admit nonempty impossible events. Although we develop most of the material on normed and inner-product spaces we need in our specific context, the appendix contains a brief review of the abstract theory.

Date: 2020
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DOI: 10.1007/978-3-030-39724-1_3

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