1 Characteristic function and covariance

1.1 Characteristic functions

Definition 1.1 Characteristic function

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The characteristic function of a measure \(\mu \) on a normed space \(E\) is the function \(E^* \to \mathbb {C}\) defined by

\begin{align*} \hat{\mu }(L) = \int _E e^{i L(x)} \: d\mu (x) \: . \end{align*}

Theorem 1.2

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In a separable Banach space, if two finite measures have same characteristic function, they are equal.

Proof ▶

Definition 1.3 Characteristic function

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The characteristic function of a measure \(\mu \) on an inner product space \(E\) is the function \(E \to \mathbb {C}\) defined by

\begin{align*} \hat{\mu }(t) = \int _E e^{i \langle t, x \rangle } \: d\mu (x) \: . \end{align*}

This is equal to the normed space version of the characteristic function applied to the linear map \(x \mapsto \langle t, x \rangle \).

Theorem 1.4

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In a separable Hilbert space, if two finite measures have same characteristic function, they are equal.

Proof ▶

Lemma 1.5

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Let \(\mu \) be a measure on \(F\) and let \(L \in F^*\). Then

\begin{align*} \widehat{L_*\mu }(x) & = \hat{\mu }(x \cdot L) \: . \end{align*}

Proof ▶

Lemma 1.6

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Let \(\mu \) be a measure on a normed space \(E\) and let \(L\) be a continuous linear map from \(E\) to \(F\). Then for all \(L' \in F^*\),

\begin{align*} \widehat{L_*\mu }(L’) = \hat{\mu }(L’ \circ L) \: . \end{align*}

Proof ▶

1.2 Covariance

Let \(F\) be a Banach space and \(E\) be a Hilbert space.

Definition 1.7 Covariance

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The covariance bilinear form of a measure \(\mu \) on \(F\) with finite second moment is the continuous bilinear form \(C_\mu : F^* \times F^* \to \mathbb {R}\) with

\begin{align*} C_\mu (L_1, L_2) & = \int _x (L_1(x) - L_1(m_\mu )) (L_2(x) - L_2(m_\mu )) \: d\mu (x) \\ & = \int _x L_1(x - m_\mu ) L_2(x- m_\mu ) \: d\mu (x) \: . \end{align*}

Lemma 1.8

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For \(\mu \) a measure on \(F\) with finite second moment and \(L \in F^*\), \(C_\mu (L, L) = \mathbb {V}_\mu [L]\).

Proof ▶

Definition 1.9 Covariance in a Hilbert space

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The covariance bilinear form of a finite measure \(\mu \) with finite second moment on a Hilbert space \(E\) is the continuous bilinear form \(C_\mu : E \times E \to \mathbb {R}\) with

\begin{align*} C’_\mu (x, y) = \int _z \langle x, z - m_\mu \rangle \langle y, z - m_\mu \rangle \: d\mu (z) \: . \end{align*}

This is \(C_\mu \) applied to the linear maps \(L_x, L_y \in E^*\) defined by \(L_x(z) = \langle x, z \rangle \) and \(L_y(z) = \langle y, z \rangle \).

Lemma 1.10

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Let \(E\) and \(F\) be two Hilbert spaces with \(F\) finite dimensional, \(\mu \) a finite measure on \(E\) with finite second moment, and \(L : E \to F\) a continuous linear map. Then the covariance bilinear form of the measure \(L_*\mu \) is given by

\begin{align*} C’_{L_*\mu }(u, v) & = C’_\mu (L^\dagger (u), L^\dagger (v)) \: , \end{align*}

in which \(L^\dagger : F \to E\) is the adjoint of \(L\).

Proof ▶

\begin{align*} C’_{L_*\mu }(u, v) & = (L_*\mu )\left[\langle u, x - m_{L_*\mu }\rangle \langle x - m_{L_*\mu }, v \rangle \right] \\ & = \mu \left[\langle u, L(x) - L(m_\mu )\rangle \langle L(x) - L(m_\mu ), v \rangle \right] \\ & = \mu \left[\langle L^\dagger (u), x - m_\mu \rangle \langle x - m_\mu , L^\dagger (v) \rangle \right] \\ & = C’_\mu (L^\dagger (u), L^\dagger (v)) \: . \end{align*}

Definition 1.11 Covariance matrix

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The covariance matrix of a finite measure \(\mu \) with finite second moment on a finite dimensional inner product space \(E\) is the positive semidefinite matrix \(\Sigma _\mu \) such that for \(u, v \in E\),

\begin{align*} \langle u, \Sigma _\mu v\rangle = \mu [\langle u, x - m_\mu \rangle \langle x - m_\mu , v \rangle ] \: . \end{align*}

This is the covariance bilinear form \(C'_\mu (u, v)\), as a matrix.

Lemma 1.12

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Let \(E\) and \(F\) be two finite dimensional inner product spaces, \(\mu \) a measure on \(E\) with finite second moment, and \(L : E \to F\) a continuous linear map. Then the covariance matrix of the measure \(L_*\mu \) has entries

\begin{align*} \langle e_i, \Sigma _{L_*\mu } e_j\rangle & = \langle L^\dagger (e_i), \Sigma _\mu L^\dagger (e_j)\rangle \: , \end{align*}

in which \(L^\dagger : F \to E\) is the adjoint of \(L\).

Proof ▶

On the left-hand side we have

\[ \langle e_i, \Sigma _{L_*\mu } e_j\rangle = C'_{L_*\mu }(e_i, e_j) = C'_\mu (L^\dagger (e_i), L^\dagger (e_j)), \]

where the last equality comes from Lemma 1.10. On the right-hand side we have

\[ \langle L^\dagger (e_i), \Sigma _{L_*\mu } L^\dagger (e_j)\rangle = C'_\mu (L^\dagger (e_i), L^\dagger (e_j)), \]

which concludes the proof.