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 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.
is typically understood to utilize the smallest
-algebra on B for which every bounded linear functional is measurable.
An equivalent definition, in this case, to the above, is that a map
is a random variable for every bounded linear functional f, or, equivalently, that
often describes some numerical quantity of a given event.
is finite or countably infinite, the random variable is called a discrete random variable[3] and its distribution can be described by a probability mass function which assigns a probability to each value in the image of
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.
(or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space
Many important properties of physical systems can be represented mathematically as matrix problems.
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.
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.
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).
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
An MRF exhibits the Markovian property where
is a set of neighbours of the random variable Xi.
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 Xi.
It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag in 1974.
[5][6] Let X be a complete separable metric space and
A Borel measure μ on X is boundedly finite if μ(A) < ∞ for every bounded Borel set A.
be the space of all boundedly finite measures on
be a complete separable metric space.
is also а complete separable metric space.
The corresponding open subsets generate a σ-algebra on
A random compact set is а measurable function
Put another way, a random compact set is a measurable function
is almost surely compact and is a measurable function for every