Gaussian measure

In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space ${\displaystyle R^{n}}$, 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 ${\displaystyle X}$ is obtained by summing a large number ${\displaystyle N}$ of independent random variables with variance 1, then ${\displaystyle X}$ has variance ${\displaystyle N}$ and its law is approximately Gaussian.

Definitions

Let ${\displaystyle n\in N}$ and let ${\displaystyle B_{0}(\mathbb {R} ^{n})}$ denote the completion of the Borel ${\displaystyle \sigma }$-algebra on ${\displaystyle \mathbb {R} ^{n}}$. Let ${\displaystyle \lambda ^{n}:B_{0}(\mathbb {R} ^{n})\to [0,+\infty ]}$ denote the usual ${\displaystyle n}$-dimensional Lebesgue measure. Then the standard Gaussian measure${\displaystyle \gamma ^{n}:B_{0}(\mathbb {R} ^{n})\to [0,1]}$ is defined by $${\displaystyle \gamma ^{n}(A)={\frac {1}{{\sqrt {2\pi }}^{n}}}\int _{A}\exp \left(-{\frac {1}{2}}\left\|x\right\|_{\mathbb {R} ^{n}}^{2}\right)\,\mathrm {d} \lambda ^{n}(x)}$$ for any measurable set ${\displaystyle A\in B_{0}(\mathbb {R} ^{n})}$. In terms of the Radon–Nikodym derivative, $${\displaystyle {\frac {\mathrm {d} \gamma ^{n}}{\mathrm {d} \lambda ^{n}}}(x)={\frac {1}{{\sqrt {2\pi }}^{n}}}\exp \left(-{\frac {1}{2}}\left\|x\right\|_{\mathbb {R} ^{n}}^{2}\right).}$$

More generally, the Gaussian measure with mean ${\displaystyle \mu \in \mathbb {R} ^{n}}$ and variance ${\displaystyle \sigma ^{2}>0}$ is given by $${\displaystyle \gamma _{\mu ,\sigma ^{2}}^{n}(A):={\frac {1}{{\sqrt {2\pi \sigma ^{2}}}^{n}}}\int _{A}\exp \left(-{\frac {1}{2\sigma ^{2}}}\left\|x-\mu \right\|_{\mathbb {R} ^{n}}^{2}\right)\,\mathrm {d} \lambda ^{n}(x).}$$

Gaussian measures with mean ${\displaystyle \mu =0}$ are known as centered Gaussian measures.

The Dirac measure ${\displaystyle \delta _{\mu }}$ is the weak limit of ${\displaystyle \gamma _{\mu ,\sigma ^{2}}^{n}}$ as ${\displaystyle \sigma \to 0}$, 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 ${\displaystyle \gamma ^{n}}$ on ${\displaystyle \mathbb {R} ^{n}}$

• 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: ${\displaystyle \lambda ^{n}\ll \gamma ^{n}\ll \lambda ^{n}}$, where ${\displaystyle \ll }$ stands for absolute continuity of measures;
• is supported on all of Euclidean space: ${\displaystyle \operatorname {supp} (\gamma ^{n})=\mathbb {R} ^{n}}$;
• is a probability measure ${\displaystyle (\gamma ^{n}(\mathbb {R} ^{n})=1)}$, and so it is locally finite;
• is strictly positive: every non-empty open set has positive measure;
• is inner regular: for all Borel sets ${\displaystyle A}$, $${\displaystyle \gamma ^{n}(A)=\sup\{\gamma ^{n}(K)\mid K\subseteq A,K{\text{ is compact}}\},}$$ so Gaussian measure is a Radon measure;
• is not translation-invariant, but does satisfy the relation $${\displaystyle {\frac {\mathrm {d} (T_{h})_{*}(\gamma ^{n})}{\mathrm {d} \gamma ^{n}}}(x)=\exp \left(\langle h,x\rangle _{\mathbb {R} ^{n}}-{\frac {1}{2}}\|h\|_{\mathbb {R} ^{n}}^{2}\right),}$$ where the derivative on the left-hand side is the Radon–Nikodym derivative, and ${\displaystyle (T_{h})_{*}(\gamma ^{n})}$ is the push forward of standard Gaussian measure by the translation map ${\displaystyle T_{h}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$, ${\displaystyle T_{h}(x)=x+h}$;
• is the probability measure associated to a normal probability distribution: $${\displaystyle Z\sim \operatorname {Normal} (\mu ,\sigma ^{2})\implies \mathbb {P} (Z\in A)=\gamma _{\mu ,\sigma ^{2}}^{n}(A).}$$

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 ${\displaystyle \gamma }$ on a separable Banach space ${\displaystyle E}$ is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional ${\displaystyle L\in E^{*}}$ except ${\displaystyle L=0}$, the push-forward measure ${\displaystyle L_{*}(\gamma )}$ is a non-degenerate (centered) Gaussian measure on ${\displaystyle \mathbb {R} }$ in the sense defined above.

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

See also

• Besov measure  – Generalization of the Gaussian measure using the Besov norm
• Cameron–Martin theorem  – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
• Covariance operator  – Operator in probability theory
• Feldman–Hájek theorem  – Theory in probability theory

References

• Bogachev, Vladimir (1998). Gaussian Measures. American Mathematical Society. ISBN   978-1470418694.
• Stroock, Daniel (2010). "Gaussian Measures on a Banach Space". Probability Theory: An Analytic View. Cambridge University Press. pp. 264–303. doi:10.1017/9781009549035.010. ISBN   978-0521132503.