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Random Variables: Topology and Geometry

In: Market-Consistent Prices

Author

Listed:
  • Pablo Koch-Medina

    (University of Zurich, Department of Banking and Finance)

  • Cosimo Munari

    (University of Zurich, Department of Banking and Finance)

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.

Suggested Citation

  • Pablo Koch-Medina & Cosimo Munari, 2020. "Random Variables: Topology and Geometry," Springer Books, in: Market-Consistent Prices, chapter 3, pages 59-82, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-39724-1_3
    DOI: 10.1007/978-3-030-39724-1_3
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