Basic Ideas

  • Sergey V. Lototsky
  • Boris L. Rozovsky
Part of the Universitext book series (UTX)


Given a probability space \((\varOmega,\mathcal{F}, \mathbb{P})\) and two measurable spaces, \((A,\mathcal{A})\) and \((B,\mathcal{B})\), a random function X is a (measurable) mapping from \((A\times \varOmega,\mathcal{A}\times \mathcal{F})\) to \((B,\mathcal{Y})\). In the traditional terminology, the random process corresponds to \(A,B \subset \mathbb{R}\); a random field corresponds to \(A = \mathbb{R}^{\mathrm{d}}\), \(B = \mathbb{R}\). A sample path , or sample trajectory of X is the function X(⋅ , ω) for fixed ωΩ. A modification of X is a random function \(\bar{X}\) such that, for every aA, \(\mathbb{P}\big(X(a) =\bar{ X}(a)\big) = 1\); note that this, in general, DOES NOT mean that \(\mathbb{P}\big(X(a) =\bar{ X}(a)\ \text{for all}\ a \in A\big) = 1\) (although it does if both X and \(\bar{X}\) are continuous).


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sergey V. Lototsky
    • 1
  • Boris L. Rozovsky
    • 2
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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