Tightness of measures

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 Hausdorff space, and let $\Sigma$ be a σ-algebra on $X$ that contains the topology $T$ . (Thus, every open subset of $X$ is a measurable set and $\Sigma$ 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 $\Sigma$ . The collection $M$ is called tight (or sometimes uniformly tight) if, for any $\varepsilon >0$ , there is a compact subset $K_{\varepsilon }$ of $X$ such that, for all measures $\mu \in M$ ,

|\mu |(X\setminus K_{\varepsilon })<\varepsilon .

where $|\mu |$ 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_{\varepsilon })>1-\varepsilon .\,

If a tight collection $M$ consists of a single measure $\mu$ , then (depending upon the author) $\mu$ 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.

Another equivalent criterion of the tightness of a collection $M$ is sequentially weakly compact. We say the family $M$ of probability measures is sequentially weakly compact if for every sequence $\left\{\mu _{n}\right\}$ from the family, there is a subsequence of measures that converges weakly to some probability measure $\mu$ . It can be shown that a family of measure is tight if and only if it is sequentially weakly compact.

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 $\mathbb {R}$ with its usual Borel topology. Let $\delta _{x}$ denote the Dirac measure, a unit mass at the point $x$ in $\mathbb {R}$ . The collection

M_{1}:=\{\delta _{n}|n\in \mathbb {N} \}

is not tight, since the compact subsets of $\mathbb {R}$ are precisely the closed and bounded subsets, and any such set, since it is bounded, has $\delta _{n}$ -measure zero for large enough $n$ . On the other hand, the collection

M_{2}:=\{\delta _{1/n}|n\in \mathbb {N} \}

is tight: the compact interval $[0,1]$ will work as $K_{\varepsilon }$ for any $\varepsilon >0$ . In general, a collection of Dirac delta measures on $\mathbb {R} ^{n}$ is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measures

Consider $n$ -dimensional Euclidean space $\mathbb {R} ^{n}$ with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

\Gamma =\{\gamma _{i}|i\in I\},

where the measure $\gamma _{i}$ has expected value (mean) $m_{i}\in \mathbb {R} ^{n}$ and covariance matrix $C_{i}\in \mathbb {R} ^{n\times n}$ . Then the collection $\Gamma$ is tight if, and only if, the collections $\{m_{i}|i\in I\}\subseteq \mathbb {R} ^{n}$ and $\{C_{i}|i\in I\}\subseteq \mathbb {R} ^{n\times n}$ 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 $(\mu _{\delta })_{\delta >0}$ on a Hausdorff topological space $X$ is said to be exponentially tight if, for any $\varepsilon >0$ , there is a compact subset $K_{\varepsilon }$ of $X$ such that

\limsup _{\delta \downarrow 0}\delta \log \mu _{\delta }(X\setminus K_{\varepsilon })<-\varepsilon .

References

Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.
Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR1102015 (See chapter 2)