Normal-inverse Gaussian distribution

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

Definitions

Let (X, T) be a topological space, 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 K_ε of X such that, for all measures μ in M,

$\left|\mu \right|(X\setminus K_{\epsilon })<\epsilon .$

where $\left|\mu \right|$ is the total variation measure of $\mu$. Very often, the measures in question are probability measures, so the last part can be written as

$\mu (K_{\epsilon })>1-\epsilon .$

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.

Examples

Compact spaces

If X is a metrisable compact space, then every collection of (possibly complex) measures on X is tight. This is not necessarily so for non-metrisable compact spaces. If we take $[0,{\omega }_1]$ with its order topology, then there exists a measure $\mu$ on it that is not inner regular. Therefore the singleton $\{\mu \}$ is not tight.

Polish spaces

If X is a Polish space, 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.

A collection of point masses

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

$M_1:=\{{\delta }_n|n\in \mathbb{\mathbb{N}}\}$

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 δ_n-measure zero for large enough n. On the other hand, the collection

$M_2:=\{{\delta }_{1/n}|n\in \mathbb{\mathbb{N}}\}$

is tight: the compact interval [0, 1] will work as K_η 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.

A collection of Gaussian measures

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

$\mathrm{\Gamma }=\{{\gamma }_i|i\in I\},$

where the measure γ_i has expected value (mean) μ_i in Rⁿ and variance σ_i² > 0. Then the collection Γ is tight if, and only if, the collections $\{{\mu }_i|i\in I\}\subseteq {\mathbb{ℝ}}^n$ and $\{{\sigma }_i^2|i\in I\}\subseteq \mathbb{ℝ}$ 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

• Finite-dimensional distribution
• Prokhorov's theorem
• 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 theory. A family of probability measures (μ_δ)_δ>0 on a Hausdorff topological space X is said to be exponentially tight if, for any η > 0, there is a compact subset K_η of X such that

${\mathrm{lim\; sup}}_{\delta \downarrow 0}\delta \log {\mu }_{\delta }(X\setminus K_{\eta })<-\eta .$

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MathSciNet (See chapter 2)