Tightness of measures

Last updated

In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity."

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Infinity mathematical concept of something without any limit

Infinity is a concept describing something without any bound, or something larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. This idea is also at the basis of infinitesimal calculus.

Contents

Definitions

Let be a topological space, and let be a σ-algebra on that contains the topology . (Thus, every open subset of is a measurable set and is at least as fine as the Borel σ-algebra on .) Let be a collection of (possibly signed or complex) measures defined on . The collection is called tight (or sometimes uniformly tight) if, for any , there is a compact subset of such that, for all measures ,

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

Open set in topology, set that does not contain any of its boundary points

In mathematics, and more specifically in topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points ; however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. These conditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, every set can be open, or no set can be open but the space itself and the empty set.

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.

where is the total variation measure of . Very often, the measures in question are probability measures, so the last part can be written as

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. The difference between a probability measure and the more general notion of measure is that a probability measure must assign value 1 to the entire probability space.

If a tight collection consists of a single measure , then (depending upon the author) may either be said to be a tight measure or to be an 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.

If is an -valued random variable whose probability distribution on is a tight measure then is said to be a separable random variable or a Radon random variable.

Random variable variable whose possible values are numerical outcomes of a random phenomenon

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon. More specifically, a random variable is defined as a function that maps the outcomes of an unpredictable process to numerical quantities, typically real numbers. It is a variable, in the sense that it depends on the outcome of an underlying process providing the input to this function, and it is random in the sense that the underlying process is assumed to be random.

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. Examples of random phenomena can include the results of an experiment or survey.

Examples

Compact spaces

If is a metrisable compact space, then every collection of (possibly complex) measures on is tight. This is not necessarily so for non-metrisable compact spaces. If we take with its order topology, then there exists a measure on it that is not inner regular. Therefore, the singleton is not tight.

Compact space Topological notions of all points being "close"

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

Polish spaces

If is a compact Polish space, then every probability measure on is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on is tight if and only if it is precompact in the topology of weak convergence.

A collection of point masses

Consider the real line with its usual Borel topology. Let denote the Dirac measure, a unit mass at the point in . The collection

is not tight, since the compact subsets of are precisely the closed and bounded subsets, and any such set, since it is bounded, has -measure zero for large enough . On the other hand, the collection

is tight: the compact interval will work as for any . In general, a collection of Dirac delta measures on is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measures

Consider -dimensional Euclidean space with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

where the measure has expected value (mean) and covariance matrix . Then the collection 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 dimension. See

Exponential tightness

A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures on a Hausdorff topological space is said to be exponentially tight if, for any , there is a compact subset of such that

Related Research Articles

In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets. Some authors require additional restrictions on the measure, as described below.

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

Fokker–Planck equation

In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck, and is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered the concept in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation, and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is known in statistical mechanics as the Liouville equation. The Fokker–Planck equation is obtained from the Master equation through Kramers–Moyal expansion.

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations.

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure.

In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis.

In probability theory, the Chernoff bound, named after Herman Chernoff but due to Herman Rubin, gives exponentially decreasing bounds on tail distributions of sums of independent random variables. It is a sharper bound than the known first- or second-moment-based tail bounds such as Markov's inequality or Chebyshev's inequality, which only yield power-law bounds on tail decay. However, the Chernoff bound requires that the variates be independent – a condition that neither Markov's inequality nor Chebyshev's inequality require, although Chebyshev's inequality does require the variates to be pairwise independent.

In the mathematical field of real analysis, Lusin's theorem states that every measurable function is a continuous function on nearly all its domain. In the informal formulation of J. E. Littlewood, "every measurable function is nearly continuous".

In measure theory Prokhorov’s theorem relates tightness of measures to relative compactness in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements.

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". It is defined to be the largest (closed) subset of X for which every open neighbourhood of every point of the set has positive measure.

In mathematics, the Lévy–Prokhorov metric is a metric on the collection of probability measures on a given metric space. It is named after the French mathematician Paul Lévy and the Soviet mathematician Yuri Vasilyevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.

In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence in measure, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be N sufficiently large for nN to ensure the 'difference' between μn and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.

Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.

In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability.

In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.

In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are “the same” from the point of view of large deviations theory.

The exponential mechanism is a technique for designing differentially private algorithms. It was developed by Frank McSherry and Kunal Talwar. Differential privacy is a technique for releasing statistical information about a database without revealing information about its individual entries.

In mathematics, and especially topology, a Poincaré complex is an abstraction of the singular chain complex of a closed, orientable manifold.

In mathematics, a càdlàg, RCLL, or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space.

References