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".

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• 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)

• Measure theory
• Measures (measure theory)

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