Random Element

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A Random Element is a measurable real-valued function that ...



References

2015

  1. V.V. Buldygin, A.B. Kharazishvili. Geometric Aspects of Probability Theory and Mathematical Statistics. – Kluwer Academic Publishers, Dordrecht. – 2000


  • (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/random_element#Definition Retrieved:2015-5-16.
    • Let [math] (\Omega, \mathcal{F}, P) [/math] be a probability space, and [math] (E, \mathcal{E}) [/math] a measurable space. A 'random element with values in E is a function which is [math] (\mathcal{F}, \mathcal{E}) [/math] -measurable. That is, a function X such that for any [math] B\in \mathcal{E} [/math], the preimage of B lies in [math] \mathcal{F} [/math] .

      Sometimes random elements with values in [math] E [/math] are called [math] E [/math] -valued random variables.

      Note if [math] (E, \mathcal{E})=(\mathbb{R}, \mathcal{B}(\mathbb{R})) [/math] , where [math] \mathbb{R} [/math] are the real numbers, and [math] \mathcal{B}(\mathbb{R}) [/math] is its Borel σ-algebra, then the definition of random element is the classical definition of random variable.

      The definition of a random element [math] X [/math] with values in a Banach space [math] B [/math] is typically understood to utilize the smallest [math] \sigma [/math] -algebra on B for which every bounded linear functional is measurable. An equivalent definition, in this case, to the above, is that a map [math] X: \Omega \rightarrow B [/math] , from a probability space, is a random element if [math] f \circ X [/math] is a random variable for every bounded linear functional f, or, equivalently, that [math] X [/math] is weakly measurable.