Gaussian measure
Type of Borel measure
In
mathematics
,
Gaussian measure
is a
Borel measure
on finite-dimensional
Euclidean space
R
n
, 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
X
is obtained by summing a large number
N
of independent random variables of order 1, then
X
is of order
and its law is approximately Gaussian.
Definitions
Let n ∈ N and let B 0 ( R n ) denote the completion of the Borel σ -algebra on R n . Let λ n : B 0 ( R n ) → [0, +∞] denote the usual n -dimensional Lebesgue measure . Then the standard Gaussian measure γ n : B 0 ( R n ) → [0, 1] is defined by
for any measurable set A ∈ B 0 ( R n ). In terms of the Radon–Nikodym derivative ,
More generally, the Gaussian measure with mean μ ∈ R n and variance σ 2 > 0 is given by
Gaussian measures with mean μ = 0 are known as centred Gaussian measures .
The
Dirac measure
δ
μ
is the
weak limit
of
as
σ
→ 0, 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 γ n on R n
- 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);
-
is
equivalent
to Lebesgue measure:
, where
stands for absolute continuity of measures;
- is supported on all of Euclidean space: supp( γ n ) = R n ;
- is a probability measure ( γ n ( R n ) = 1), and so it is locally finite ;
- is strictly positive : every non-empty open set has positive measure;
-
is
inner regular
: for all Borel sets
A
,
-
is not
translation
-
invariant
, but does satisfy the relation
-
is the probability measure associated to a
normal
probability distribution
:
Infinite-dimensional spaces
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 γ on a separable Banach space E is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional L ∈ E ∗ except L = 0, the push-forward measure L ∗ ( γ ) is a non-degenerate (centered) Gaussian measure on R in the sense defined above.
For example, classical Wiener measure on the space of continuous paths is a Gaussian measure.
References
- Bogachev, Vladimir (1998). Gaussian Measures . American Mathematical Society. ISBN 978-1470418694 .
- Stroock, Daniel (2010). Probability Theory: An Analytic View . Cambridge University Press. ISBN 978-0521132503 .
See also
- Besov measure – mathematical term Pages displaying wikidata descriptions as a fallback - a generalisation of Gaussian measure
- Cameron–Martin theorem – Theorem of measure theory
- Covariance operator – Operator in probability theory
- Feldman–Hájek theorem
Basic concepts | ||||
---|---|---|---|---|
Sets | ||||
Types of Measures |
|
|||
Particular measures | ||||
Maps | ||||
Main results |
|
|||
Other results |
|
|||
Applications & related |
Basic concepts | |
---|---|
Derivatives | |
Measurability | |
Integrals | |
Results | |
Related | |
Functional calculus | |
Applications |