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In mathematics, tightness is a concept in measure theory, the intuitive idea being that a given collection of measures does not "escape to infinity." Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, a measure is a function that assigns a number, e. ...
The infinity symbol ∞ in several typefaces The word infinity comes from the Latin infinitas or unboundedness. ...
Definition Let (Ω,T) be a topological space, and let M be a collection of measures defined on σ(T), the smallest sigma algebra containing the topology T (so that every open set in Ω is measurable). The collection M is called tight if, for any  , there is a compact set  such that, for all measures ,  . Very often, the measures in question are probability measures, so the last part can be written as  . Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...
Examples
Compact spaces If Ω is a compact space, then every collection of probability measures on Ω is tight. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
A collection of point masses Consider the real line  with its usual Borel topology. Let δx denote the Dirac delta, a unit mass at the point  . The collection  is not tight, since the compact subsets of are precisely the closed and bounded subsets, and any such set, since 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  for any . In general, a collection of Dirac delta measures on  is tight if, and only if, the collection of their supports is bounded. In mathematics, the real line is simply the set of real numbers. ...
The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...
A collection of Gaussian measures Consider n-dimensional Euclidean space with its usual Borel topology and sigma algebra. Consider a collection of Gaussian measures  , where the measure γi has expected value (mean)  and variance  . Then the collection is tight if, and only if, the collections  and  are both bounded. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space , closely related to the normal distribution in statistics. ...
In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
In statistics, mean has two related meanings: the average in ordinary English, which is also called the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...
In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...
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 In probability theory, there exist several different notions of convergence of random variables. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
:For other senses of this word, see dimension (disambiguation). ...
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