Gaussian 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.

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, +∞] is defined by

:$gamma^\{n\}\; (A)\; =\; frac\{1\}\{sqrt\{2\; pi\}^\{n\; int\_\{A\}\; exp\; left(\; -\; frac\{1\}\{2\}\; |\; x\; |\_\{mathbb\{R\}^\{n^\{2\}\; ight)\; ,\; mathrm\{d\}\; lambda^\{n\}\; (x)$

for any measurable set "A" ∈ "B"₀(R^"n"). In terms of the Radon-Nikodym derivative,

:$frac\{mathrm\{d\}\; gamma^\{n\{mathrm\{d\}\; lambda^\{n\; (x)\; =\; frac\{1\}\{sqrt\{2\; pi\}^\{n\; exp\; left(\; -\; frac\{1\}\{2\}\; |\; x\; |\_\{mathbb\{R\}^\{n^\{2\}\; ight).$

More generally, the Gaussian measure with mean "μ" ∈ R^"n" and variance "σ"² > 0 is given by

:$gamma\_\{mu,\; sigma^\{2^\{n\}\; (A)\; :=\; frac\{1\}\{sqrt\{2\; pi\; sigma^\{2^\{n\; int\_\{A\}\; exp\; left(\; -\; frac\{1\}\{2\; sigma^\{2\; |\; x\; -\; mu\; |\_\{mathbb\{R\}^\{n^\{2\}\; ight)\; ,\; mathrm\{d\}\; lambda^\{n\}\; (x).$

Gaussian measures with mean "μ" = 0 are known as centred Gaussian measures.

The Dirac measure "δ"_"μ" is the weak limit of $gamma\_\{mu,\; sigma^\{2^\{n\}$ 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 of Gaussian measure

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: $lambda^\{n\}\; ll\; gamma^\{n\}\; ll\; lambda^\{n\}$, where $ll$ 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",

:$gamma^\{n\}\; (A)\; =\; sup\; \{\; gamma^\{n\}\; (K)\; |\; K\; subseteq\; A,\; K\; mbox\{\; is\; compact\}\; \},$

so Gaussian measure is a Radon measure;
* is not translation-invariant, but does satisfy the relation

:$frac\{mathrm\{d\}\; (T\_\{h\})\_\{*\}\; (gamma^\{n\})\}\{mathrm\{d\}\; gamma^\{n\; (x)\; =\; exp\; left(\; langle\; h,\; x\; angle\_\{mathbb\{R\}^\{n\; -\; frac\{1\}\{2\}\; |\; h\; |\_\{mathbb\{R\}^\{2^\{2\}\; ight),$

:where the derivative on the left-hand side is the Radon-Nikodym derivative, and ("T"_"h")_∗("γ"^"n") is the push forward of standard Gaussian measure by the translation map "T"_"h" : R^"n" → R^"n", "T"_"h"("x") = "x" + "h";
* is the probability measure associated to a normal probability distribution:

:$Z\; sim\; mathrm\{Normal\}\; (mu,\; sigma^\{2\})\; implies\; mathbb\{P\}\; (Z\; in\; A)\; =\; gamma\_\{mu,\; sigma^\{2^\{n\}\; (A).$

Gaussian measures on 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 infinte-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 (centred) Gaussian measure if, for every linear functional "L" ∈ "E"^∗ except "L" = 0, the push-forward measure "L"_∗("γ") is a non-degenerate (centred) Gaussian measure on R in the sense defined above.

For example, classical Wiener measure on the space of continuous paths is a Gaussian measure.

ee also

* Cameron-Martin theorem