In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space
, closely related to the normal distribution in statistics.
There is also a generalization to infinite-dimensional spaces.
Gaussian measures are named after the German mathematician Carl Friedrich Gauss.
One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem.
Loosely speaking, it states that if a random variable
is obtained by summing a large number
of independent random variables with variance 1, then
and its law is approximately Gaussian.
denote the completion of the Borel
denote the usual
-dimensional Lebesgue measure.
Then the standard Gaussian measure
for any measurable set
More generally, the Gaussian measure with mean
Gaussian measures with mean
are known as centered Gaussian measures.
The Dirac measure
is the weak limit of
, and is considered to be a degenerate Gaussian measure; in contrast, Gaussian measures with finite, non-zero variance are called non-degenerate Gaussian measures.
The standard Gaussian measure
It can be shown that there is no analogue of Lebesgue measure on an infinite-dimensional vector space.
Even so, it is possible to define Gaussian measures on infinite-dimensional spaces, the main example being the abstract Wiener space construction.
A Borel measure
on a separable Banach space
is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional
, the push-forward measure
is a non-degenerate (centered) Gaussian measure on
in the sense defined above.
For example, classical Wiener measure on the space of continuous paths is a Gaussian measure.