Random element

The term random element was introduced by Maurice Frechet in 1948 to refer to a random variable that takes values in spaces more general than had previously been widely considered. Frechet commented that the "development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experience can be described by number or a finite set of numbers, to schemes where outcomes of experience represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets".

The modern day usage of "random element" frequently assumes the space of values is a topological vector space, often a Banach or Hilbert space with a specified natural sigma algebra of subsets.

Definition

Let $(Omega,\; mathcal\{F\},\; mathbf\{P\})$ be a probability space, and $(E,mathcal\{E\})$ be a measurable space. They say, that function $X\; :\; (Omega,\; mathcal\{F\},\; mathbf\{P\})\; o\; (E,mathcal\{E\})$ is $(mathcal\{F\},\; mathcal\{E\})$-measurable function, or random element (with values in $E$), if for any $B\; in\; mathcal\{E\}$

: $\{omega\; :\; X(omega)\; in\; B\; \}\; in\; mathcal\{F\}.$

Sometimes random elements (with values in $E$) are called also $E$-valued random variables.

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

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

Random elements of the various nature

* Random variable
* Discrete random variable
* Continuous random variable
* Complex random variable
* Simple random variable
* Random vector
* Random matrix
* Random function
* Random process
* Random field
* Random measure
* Random set
* Random closed set
* Random compact set
* Random “point”
* Random figureref|Stoyan
* Random shaperef|Stoyan
* Random finite set
* Random finite abstract set
* Random set of events

References

* [http://www.numdam.org/item?id=AIHP_1948__10_4_215_0 1 Frechet, M. (1948) Les elements aleatories de nature quelconque dans un espace distancie. Ann.Inst.H.Poincare 10, 215-310.
* Prokhorov Yu.V. (1999) Random element. Probability and Mathematical statistics. Encyclopedia. Moscow: "Great Russian Encyclopedia", P.623.
* Mourier E. (1955) Elements aleatoires dans un espace de Banach (These). Paris.
* Hoffman-Jorgensen J., Pisier G. (1976) "Ann.Probab.", v.4, 587-589.
* Stoyan D., and H.Stoyan (1994) Fractals, Random Shapes and Point Fields. Methods of Geometrical Statistics. Chichester, New York: John Wiley & Sons. ISBN 0-471-93757-6

External links

* [http://eom.springer.de/R/r077280.htm Entry in Springer Encyclopedia of Mathematics]