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 who 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 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.”[1]
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.[2]
Let
(\Omega,l{F},P)
(E,l{E})
(l{F},l{E})
B\inl{E}
l{F}
Sometimes random elements with values in
E
E
Note if
(E,l{E})=(R,l{B}(R))
R
l{B}(R)
The definition of a random element
X
B
\sigma
X:\Omega → B
f\circX
X
See main article: Random variable. A random variable is the simplest type of random element. It is a map
X\colon\Omega\toR
\Omega
R
As a real-valued function,
X
When the image (or range) of
X
X
X
X=(X1,...,X
| T | |
| n) |
(\Omega,l{F},P)
\Omega
l{F}
P
Random vectors are often used as the underlying implementation of various types of aggregate random variables, e.g. a random matrix, random tree, random sequence, random process, etc.
See main article: Random matrix theory. A random matrix is a matrix-valued random element. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.
See main article: Random function. A random function is a type of random element in which a single outcome is selected from some family of functions, where the family consists some class of all maps from the domain to the codomain. For example, the class may be restricted to all continuous functions or to all step functions. The values determined by a random function evaluated at different points from the same realization would not generally be statistically independent but, depending on the model, values determined at the same or different points from different realisations might well be treated as independent.
See main article: Random process. A Random process is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
In the simple case of discrete time, as opposed to continuous time, a stochastic process involves a sequence of random variables and the time series associated with these random variables (for example, see Markov chain, also known as discrete-time Markov chain).
(\Omega,l{F},P)
\{Ft:t\inT\}
Ft
Several kinds of random fields exist, among them the Markov random field (MRF), Gibbs random field (GRF), conditional random field (CRF), and Gaussian random field. An MRF exhibits the Markovian property
P(Xi=xi|Xj=xj,i ≠ j)=P(Xi=xi|\partiali),
where
\partiali
P(Xi=xi|\partiali)=
| P(\omega) | |
| \sum\omega'P(\omega') |
,
where Ω' is the same realization of Ω, except for random variable Xi. It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag in 1974.
See main article: Random measure. A random measure is a measure-valued random element.[5] [6] Let X be a complete separable metric space and
ak{B}(X)
MX
ak{B}(X)
\mu=\mud+\mua=\mud+
| N | |
| \sum | |
| n=1 |
\kappan
| \delta | |
| Xn |
,
Here
\mud
\mua
A random set is a set-valued random element.
One specific example is a random compact set. Let
(M,d)
l{K}
M
h
l{K}
h(K1,K2):=max\left\{
| \sup | |
| a\inK1 |
| inf | |
| b\inK2 |
d(a,b),
| \sup | |
| b\inK2 |
| inf | |
| a\inK1 |
d(a,b)\right\}.
(l{K},h)
l{K}
l{B}(l{K})
l{K}
K
(\Omega,l{F},P)
(l{K},l{B}(l{K}))
Put another way, a random compact set is a measurable function
K\colon\Omega\to2M
K(\omega)
\omega\mapstoinfbd(x,b)
is a measurable function for every
x\inM
These include random points, random figures,[8] and random shapes.[8]