Tightness of measures

Template:Short description Template:No footnotes 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 $\mathrm{\Sigma }$ be a σ-algebra on $X$ that contains the topology $T$. (Thus, every open subset of $X$ is a measurable set and $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$. The collection $M$ is called tight (or sometimes uniformly tight) if, for any $\epsilon >0$, there is a compact subset $K_{\epsilon }$ of $X$ such that, for all measures $\mu \in M$,

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

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

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

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 metrizable 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 ${\delta }_x$ denote the Dirac measure, a unit mass at the point $x$ in $ℝ$. The collection $$M_1:=\{{\delta }_n\mid n\in \mathbb{N}\}$$ 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 ${\delta }_n$-measure zero for large enough $n$. On the other hand, the collection $$M_2:=\{{\delta }_{1/n}\mid n\in \mathbb{N}\}$$ is tight: the compact interval $[0,1]$ will work as $K_{\epsilon }$ for any $\epsilon >0$. In general, a collection of Dirac delta measures on ${ℝ}^n$ is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measures

Consider $n$-dimensional Euclidean space ${ℝ}^n$ with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures $$\mathrm{\Gamma }=\{{\gamma }_i\mid i\in I\},$$ where the measure ${\gamma }_i$ has expected value (mean) $m_i\in {ℝ}^n$ and covariance matrix $C_i\in {ℝ}^{n\times n}$. Then the collection $\mathrm{\Gamma }$ is tight if, and only if, the collections $\{m_i\mid i\in I\}\subseteq {ℝ}^n$ and $\{C_i\mid i\in I\}\subseteq {ℝ}^{n\times n}$ 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

• Finite-dimensional distribution
• Prokhorov's theorem
• Lévy–Prokhorov metric
• Weak convergence of measures
• Tightness in classical Wiener space
• Tightness in Skorokhod space

Tightness and stochastic ordering

A family of real-valued random variables $\{X_i{\}}_{i\in I}$ is tight if and only if there exists an almost surely finite random variable $X$ such that $|X_i|{\le }_{\mathrm{s}\mathrm{t}}X$ for all $i\in I$, where ${\le }_{\mathrm{s}\mathrm{t}}$ denotes the stochastic order defined by $A{\le }_{\mathrm{s}\mathrm{t}}B$ if $E\varphi (A)\le E\varphi (B)$ for all nondecreasing functions $\varphi$.

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 $\epsilon >0$, there is a compact subset $K_{\epsilon }$ of $X$ such that

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

References

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