Tightness of measures
Encyclopedia
In mathematics
, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity
."
, and let Σ be a σ-algebra on X that contains the topology T. (Thus, every open subset
of X is a measurable set and Σ is at least as fine as the Borel σ-algebra on X.) Let M be a collection of (possibly signed
or complex
) measures defined on Σ. The collection M is called tight (or sometimes uniformly tight) if, for any ε > 0, there is a compact subset
Kε of X such that, for all measures μ in M,
where
is the variation measure of 
. Very often, the measures in question are probability measure
s, so the last part can be written as
If a tight collection M consists of a single measure μ, then (depending upon the author) μ may either be said to be a tight measure or to be an inner regular measure
.
If Y is an X-valued random variable
whose probability distribution
on X is a tight measure then Y is said to be a separable random variable or a Radon random variable.
, then every collection of (possibly complex) measures on X is tight.
, then every probability measure on X is tight. Furthermore, by Prokhorov's theorem
, a collection of probability measures on X is tight if and only if
it is precompact
in the topology of weak convergence
.
R with its usual Borel topology. Let δx denote the Dirac measure, a unit mass at the point x in R. The collection
is not tight, since the compact subsets of R are precisely the closed
and bounded
subsets, and any such set, since it is bounded, has δn-measure zero for large enough n. On the other hand, the collection
is tight: the compact interval [0, 1] will work as Kη for any η > 0. In general, a collection of Dirac delta measures on Rn is tight if, and only if, the collection of their supports
is bounded.
Rn with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measure
s
where the measure γi has expected value
(mean
) μi in Rn and variance
σi2 > 0. Then the collection Γ is tight if, and only if, the collections
and 
are both bounded.
. See
. A family of probability measure
s (μδ)δ>0 on a Hausdorff
topological space X is said to be exponentially tight if, for any η > 0, there is a compact subset Kη of X 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...
, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
."
Definitions
Let (X, T) be 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...
, and let Σ be a σ-algebra on X that contains the topology T. (Thus, every open subset
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
of X is a measurable set and Σ is at least as fine as the Borel σ-algebra on X.) Let M be a collection of (possibly signed
Signed measure
In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...
or complex
Complex measure
In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size is a complex number.-Definition:...
) measures defined on Σ. The collection M is called tight (or sometimes uniformly tight) if, for any ε > 0, there is a compact subset
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Kε of X such that, for all measures μ in M,
where
is the variation measure of . Very often, the measures in question are probability measureProbability 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, so the last part can be written as
If a tight collection M consists of a single measure μ, then (depending upon the author) μ may either be said to be a tight measure or to be an inner regular measure
Inner regular measure
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.-Definition:...
.
If Y is an X-valued 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...
whose probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
on X is a tight measure then Y is said to be a separable random variable or a Radon random variable.
Compact spaces
If X is a compact spaceCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, then every collection of (possibly complex) measures on X is tight.
Polish spaces
If X is a Polish spacePolish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
, then every probability measure on X is tight. Furthermore, by Prokhorov's theorem
Prokhorov's theorem
In measure theory Prokhorov’s theorem relates tightness of measures to weak compactness in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilevich Prokhorov, who considered probability measures on complete separable metric spaces...
, a collection of probability measures on X is tight if and only if
it is precompact
Relatively compact subspace
In mathematics, a relatively compact subspace Y of a topological space X is a subset whose closure is compact....
in the topology of weak convergence
Convergence of measures
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. Three of the most common notions of convergence are described below.-Total variation convergence of measures:...
.
A collection of point masses
Consider the real lineReal line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R with its usual Borel topology. Let δx denote the Dirac measure, a unit mass at the point x in R. The collection
is not tight, since the compact subsets of R are precisely the closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
and bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
subsets, and any such set, since it is bounded, has δn-measure zero for large enough n. On the other hand, the collection
is tight: the compact interval [0, 1] will work as Kη for any η > 0. In general, a collection of Dirac delta measures on Rn is tight if, and only if, the collection of their supports
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"...
is bounded.
A collection of Gaussian measures
Consider n-dimensional Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measure
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces...
s
where the measure γi has expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
(mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
) μi in Rn and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
σi2 > 0. Then the collection Γ is tight if, and only if, the collections
and are both bounded.Tightness and convergence
Tightness is often a necessary criterion for proving the weak convergence of a sequence of probability measures, especially when the measure space has infinite dimensionDimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
. See
- Finite-dimensional distributionFinite-dimensional distributionIn mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure onto a finite-dimensional vector space .-Finite-dimensional distributions of a measure:Let be a measure space...
- Prokhorov's theoremProkhorov's theoremIn measure theory Prokhorov’s theorem relates tightness of measures to weak compactness in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilevich Prokhorov, who considered probability measures on complete separable metric spaces...
- Tightness in classical Wiener space
- Tightness in Skorokhod space
Exponential tightness
A generalization of tightness is the concept of exponential tightness, which has applications in large deviations theoryLarge deviations theory
In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. Some basic ideas of the theory can be tracked back to Laplace and Cramér, although a clear unified formal definition was introduced in 1966 by Varadhan...
. A family 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 (μδ)δ>0 on a Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological space X is said to be exponentially tight if, for any η > 0, there is a compact subset Kη of X such that