Tightness of measures explained

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\inM

,

|\mu|(X\setminusK\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\{\mun\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,\omega1]

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 ifit is precompact in the topology of weak convergence.

A collection of point masses

R

with its usual Borel topology. Let

\deltax

denote the Dirac measure, a unit mass at the point

x

in

R

. The collection

M1:=\{\deltan|n\inN\}

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

\deltan

-measure zero for large enough

n

. On the other hand, the collection

M2:=\{\delta1|n\inN\}

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

Rn

is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measures

Consider

n

-dimensional Euclidean space

Rn

with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

\Gamma=\{\gammai|i\inI\},

where the measure

\gammai

has expected value (mean)

mi\inRn

and covariance matrix

Ci\inRn

. Then the collection

\Gamma

is tight if, and only if, the collections

\{mi|i\inI\}\subseteqRn

and

\{Ci|i\inI\}\subseteqRn

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

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\deltalog\mu\delta(X\setminusK\varepsilon)<-\varepsilon.

References