5 Gaussian measures

Definition 5.1

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A measure on $R$ is Gaussian if it is equal to $N (m, σ^{2})$ for some $m \in R$ and $σ^{2} \geq 0$ , in which $N (m, σ^{2})$ is the measure absolutely continuous with respect to the Lebesgue measure with density $x \mapsto \frac{1}{\sqrt{2 π σ^{2}}} e^{- \frac{1}{2 σ^{2}} (x - m)^{2}}$ if $σ^{2} > 0$ and the Dirac probability measure at $m$ if $σ^{2} = 0$ .

Lemma 5.2

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A real Gaussian measure is a probability measure.

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Lemma 5.3

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The Gaussian distribution $N (m, σ^{2})$ has characteristic function $ϕ (t) = e^{i t m - σ^{2} t^{2} / 2}$ .

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Lemma 5.4

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The sum of two independent real Gaussian random variables is Gaussian.

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Definition 5.5

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A Borel measure $μ$ on a separable Banach space $E$ is said to be Gaussian if for all continuous linear maps $ℓ : E \to R$ , the pushforward $ℓ_{*} μ$ is a real Gaussian measure.

Lemma 5.6

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A Gaussian measure is a probability measure.

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Lemma 5.7

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The real Gaussian measures are Gaussian measures in the sense of Definition 5.5.

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Lemma 5.8

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The sum of two independent Gaussian random variables is Gaussian.

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Lemma 5.9

Let $E$ be a finite dimensional real inner product space and let $b_{1}, \dots, b_{d}$ be an orthonormal basis of $E$ . Let $X_{1}, \dots, X_{d}$ be independent standard Gaussian random variables on $R$ . Then the law of $X_{1} b_{1} + \dots + X_{d} b_{d}$ is a Gaussian measure on $E$ .

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