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 topological 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.

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 compact 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

• Finite-dimensional distribution
• Prokhorov's theorem
• Lévy–Prokhorov metric
• weak convergence of measures
• 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 $(\mu_{\delta})_{\delta > 0}$ 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. MR 1102015 (See chapter 2)