Gaussian measure

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In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, 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 \sqrt{N} and its law is approximately Gaussian.

Definitions[edit]

Let nN and let B0(Rn) denote the completion of the Borel σ-algebra on Rn. Let λn : B0(Rn) → [0, +∞] denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure γn : B0(Rn) → [0, 1] is defined by

\gamma^{n} (A) = \frac{1}{\sqrt{2 \pi}^{n}} \int_{A} \exp \left( - \frac{1}{2} \| x \|_{\mathbb{R}^{n}}^{2} \right) \, \mathrm{d} \lambda^{n} (x)

for any measurable set AB0(Rn). 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} \right).

More generally, the Gaussian measure with mean μRn and variance σ2 > 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} \right) \, \mathrm{d} \lambda^{n} (x).

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

The Dirac measure δμ 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[edit]

The standard Gaussian measure γn on Rn

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

so Gaussian measure is a Radon measure;

\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 (Th)(γn) is the push forward of standard Gaussian measure by the translation map Th : RnRn, Th(x) = x + h;
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[edit]

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 LE 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.

See also[edit]