Random element

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.”
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 formula_1 be a probability space, and formula_2 a measurable space. A random element with values in "E" is a function which is formula_3-measurable. That is, a function X such that for any formula_4, the preimage of B lies in formula_5.
Sometimes random elements with values in formula_6 are called formula_6-valued random variables.
Note if formula_8, where formula_9 are the real numbers, and formula_10 is its Borel σ-algebra, then the definition of random element is the classical definition of random variable.
The definition of a random element formula_11 with values in a Banach space formula_12 is typically understood to utilize the smallest formula_13-algebra on "B" for which every bounded linear functional is measurable. An equivalent definition, in this case, to the above, is that a map formula_14, from a probability space, is a random element if formula_15 is a random variable for every bounded linear functional "f", or, equivalently, that formula_11 is weakly measurable.
Examples of random elements.
Random variable.
A "random variable" is the simplest type of random element. It is a map formula_17 is a measurable function from the set of possible outcomes formula_18 to formula_9.
As a real-valued function, formula_11 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 formula_11 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 formula_11. If the image is uncountably infinite then formula_11 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.
A random vector is a column vector formula_24 (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space formula_1, where formula_26 is the sample space, formula_5 is the sigma-algebra (the collection of all events), and formula_28 is the probability measure (a function returning each event's probability).
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.
Random matrix.
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.
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.
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).
Random field.
Given a probability space formula_1 and a measurable space X,
an "X"-valued random field is a collection of "X"-valued
random variables indexed by elements in a topological space "T". That is, a random field "F" is a collection
where each formula_31 is an "X"-valued random variable.
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
where formula_33 is a set of neighbours of the random variable "X""i". In other words, the probability that a random variable assumes a value depends on the other random variables only through the ones that are its immediate neighbours. The probability of a random variable in an MRF is given by
where Ω' is the same realization of Ω, except for random variable "X""i". It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag in 1974.
Random measure.
A random measure is a measure-valued random element. Let X be a complete separable metric space and formula_35 the σ-algebra of its Borel sets. A Borel measure μ on X is boundedly finite if μ(A) < ∞ for every bounded Borel set A. Let formula_36 be the space of all boundedly finite measures on formula_35. Let be a probability space, then a random measure maps from this probability space to the measurable space . A measure generally might be decomposed as:
Here formula_39 is a diffuse measure without atoms, while formula_40 is a purely atomic measure.
Random set.
A random set is a set-valued random element.
One specific example is a random compact set. Let formula_41 be a complete separable metric space. Let formula_42 denote the set of all compact subsets of formula_43. The Hausdorff metric formula_44 on formula_42 is defined by
formula_47 is also а complete separable metric space. The corresponding open subsets generate a σ-algebra on formula_42, the Borel sigma algebra formula_49 of formula_42.
A random compact set is а measurable function formula_51 from а probability space formula_52 into formula_53.
Put another way, a random compact set is a measurable function formula_54 such that formula_55 is almost surely compact and
is a measurable function for every formula_57.
Random geometric objects.
These include random points, random figures, and random shapes.