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 GermanmathematicianCarl 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 has variance and its law is approximately Gaussian.
Definitions
Let and let denote the completion of the Borel -algebra on . Let denote the usual -dimensional Lebesgue measure. Then the standard Gaussian measure is defined by
for any measurable set . In terms of the
Radon–Nikodym derivative
,
More generally, the Gaussian measure with mean and variance is given by
Gaussian measures with mean are known as centered Gaussian measures.
of as , and is considered to be a degenerate Gaussian measure; in contrast, Gaussian measures with finite, non-zero variance are called non-degenerate Gaussian measures.
Properties
The standard Gaussian measure on
is a Borel measure (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);
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 separableBanach space is said to be a non-degenerate (centered) Gaussian measure if, for every
linear functional
except , the
push-forward measure
is a non-degenerate (centered) Gaussian measure on <math>\mathbb{R}<\math> in the sense defined above.
Besov measure – generalization of the Gaussian measure using the Besov normPages displaying wikidata descriptions as a fallback - a generalisation of Gaussian measure
Cameron–Martin theorem – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.