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".



Let be a Hausdorff 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 ,

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

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 .

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.


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.

Polish spaces

If is a 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.

Pauli matrices Matrices important in quantum mechanics and the study of spin

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

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.

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 Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

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

Newman–Penrose formalism Notation in general relativity

The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

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, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.

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.

Classical Wiener space

In mathematics, classical Wiener space is the collection of all continuous functions on a given domain, taking values in a metric space. Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.

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 of measures, 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.

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 real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in Lp in terms of convergence in measure and a condition related to uniform integrability.

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.

In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y without any need to extend definitions from real/complex-valued functions to Y-valued functions.