In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet (1948) who commented:
[the] development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments 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 ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space, and ( E , E ) {\displaystyle (E,{\mathcal {E}})} a measurable space. A random element with values in E is a function X: Ω→E which is ( F , E ) {\displaystyle ({\mathcal {F}},{\mathcal {E}})} -measurable. That is, a function X such that for any B ∈ E {\displaystyle B\in {\mathcal {E}}} , the preimage of B lies in F {\displaystyle {\mathcal {F}}} . Sometimes random elements with values in E {\displaystyle E} are called E {\displaystyle E} -valued random variables. Note if ( E , E ) = ( R , B ( R ) ) {\displaystyle (E,{\mathcal {E}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))} , where R {\displaystyle \mathbb {R} } are the real numbers, and B ( R ) {\displaystyle {\mathcal {B}}(\mathbb {R} )} is its Borel σ-algebra, then the definition of random element is the classical definition of random variable. The definition of a random element X {\displaystyle X} with values in a Banach space B {\displaystyle B} is typically understood to utilize the smallest σ {\displaystyle \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 : Ω → B {\displaystyle X:\Omega \rightarrow B} , from a probability space, is a random element if f ∘ X {\displaystyle f\circ X} is a random variable for every bounded linear functional f, or, equivalently, that X {\displaystyle X} is weakly measurable.
Examples of random elements
Random variable
A random variable is the simplest type of random element. It is a map X : Ω → R {\displaystyle X\colon \Omega \to \mathbb {R} } is a measurable function from the set of possible outcomes Ω {\displaystyle \Omega } to R {\displaystyle \mathbb {R} } . As a real-valued function, X {\displaystyle X} often describes some numerical quantity of a given event. E.g. the number of heads after a certain number of coin flips; the heights of different people. When the image (or range) of X {\displaystyle X} is finite or countably infinite, the random variable is called a discrete random variable and its distribution can be described by a probability mass function which assigns a probability to each value in the image of X {\displaystyle X} . If the image is uncountably infinite then X {\displaystyle X} is called a continuous random variable. In the special case that it is absolutely continuous, its distribution can be described by a probability density function, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous, for example a mixture distribution. Such random variables cannot be described by a probability density or a probability mass function.
Random vector
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