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

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

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

${\displaystyle \mu (K_{\varepsilon })>1-\varepsilon .\,}$

If a tight collection ${\displaystyle M}$ consists of a single measure ${\displaystyle \mu }$, then (depending upon the author) ${\displaystyle \mu }$ may either be said to be a tight measure or to be an inner regular measure.

If ${\displaystyle Y}$ is an ${\displaystyle X}$-valued random variable whose probability distribution on ${\displaystyle X}$ is a tight measure then ${\displaystyle Y}$ is said to be a separable random variable or a Radon random variable.

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

Examples

Compact spaces

If ${\displaystyle X}$ is a metrizable compact space, then every collection of (possibly complex) measures on ${\displaystyle X}$ is tight. This is not necessarily so for non-metrisable compact spaces. If we take ${\displaystyle [0,\omega _{1}]}$ with its order topology, then there exists a measure ${\displaystyle \mu }$ on it that is not inner regular. Therefore, the singleton ${\displaystyle \{\mu \}}$ is not tight.

Polish spaces

If ${\displaystyle X}$ is a Polish space, then every finite measure on ${\displaystyle X}$ is tight; this is Ulam's theorem. Furthermore, by Prokhorov's theorem, a collection of probability measures on ${\displaystyle X}$ is tight if and only if it is precompact in the topology of weak convergence.

A collection of point masses

Consider the real line ${\displaystyle \mathbb {R} }$ with its usual Borel topology. Let ${\displaystyle \delta _{x}}$ denote the Dirac measure, a unit mass at the point ${\displaystyle x}$ in ${\displaystyle \mathbb {R} }$. The collection $${\displaystyle M_{1}:=\{\delta _{n}\mid n\in \mathbb {N} \}}$$ is not tight, since the compact subsets of ${\displaystyle \mathbb {R} }$ are precisely the closed and bounded subsets, and any such set, since it is bounded, has ${\displaystyle \delta _{n}}$-measure zero for large enough ${\displaystyle n}$. On the other hand, the collection $${\displaystyle M_{2}:=\{\delta _{1/n}\mid n\in \mathbb {N} \}}$$ is tight: the compact interval ${\displaystyle [0,1]}$ will work as ${\displaystyle K_{\varepsilon }}$ for any ${\displaystyle \varepsilon >0}$. In general, a collection of Dirac delta measures on ${\displaystyle \mathbb {R} ^{n}}$ is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measures

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

• 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 ${\displaystyle \{X_{i}\}_{i\in I}}$ is tight if and only if there exists an almost surely finite random variable ${\displaystyle X}$ such that ${\displaystyle |X_{i}|\leq _{\mathrm {st} }X}$ for all ${\displaystyle i\in I}$, where ${\displaystyle \leq _{\mathrm {st} }}$ denotes the stochastic order defined by ${\displaystyle A\leq _{\mathrm {st} }B}$ if ${\displaystyle E\phi (A)\leq E\phi (B)}$ for all nondecreasing functions ${\displaystyle \phi }$. ^[1]

Exponential tightness

A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures ${\displaystyle (\mu _{\delta })_{\delta >0}}$ on a Hausdorff topological space ${\displaystyle X}$ is said to be exponentially tight if, for any ${\displaystyle \varepsilon >0}$, there is a compact subset ${\displaystyle K_{\varepsilon }}$ of ${\displaystyle X}$ such that

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