Skorokhod's representation theorem
Encyclopedia
In mathematics
and statistics
, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence
of probability measure
s whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent
sequence of random variable
s defined on a common probability space
. It is named for the Ukrainian
mathematician
A.V. Skorokhod
.
S; suppose that μn converges weakly to some probability measure μ on S as n → ∞. Suppose also that the support
of μ is separable. Then there exist random variables Xn, X defined on a common probability space (Ω, F, P) such that
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
s whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
sequence of random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s defined on a common probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
. It is named for the Ukrainian
Ukraine
Ukraine is a country in Eastern Europe. It has an area of 603,628 km², making it the second largest contiguous country on the European continent, after Russia...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
A.V. Skorokhod
Anatoliy Skorokhod
Anatoliy Volodymyrovych Skorokhod was a Soviet and Ukrainian mathematician, and an academician of the National Academy of Sciences of Ukraine from 1985 to his death in 2011....
.
Statement of the theorem
Let μn, n ∈ N be a sequence of probability measures on a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
S; suppose that μn converges weakly to some probability measure μ on S as n → ∞. Suppose also that the support
Support (measure theory)
In mathematics, the support of a measure μ on a measurable topological space is a precise notion of where in the space X the measure "lives"...
of μ is separable. Then there exist random variables Xn, X defined on a common probability space (Ω, F, P) such that
- (Xn)∗(P) = μn (i.e. μn is the distribution/law of Xn);
- X∗(P) = μ (i.e. μ is the distribution/law of X); and
- Xn(ω) → X(ω) as n → ∞ for every ω ∈ Ω.
See also
- Convergence in distribution