1 Characteristic function and covariance
1.1 Characteristic functions
The characteristic function of a measure \(\mu \) on a normed space \(E\) is the function \(E^* \to \mathbb {C}\) defined by
In a separable Banach space, if two finite measures have same characteristic function, they are equal.
The characteristic function of a measure \(\mu \) on an inner product space \(E\) is the function \(E \to \mathbb {C}\) defined by
This is equal to the normed space version of the characteristic function applied to the linear map \(x \mapsto \langle t, x \rangle \).
In a separable Hilbert space, if two finite measures have same characteristic function, they are equal.
Let \(\mu \) be a measure on \(F\) and let \(L \in F^*\). Then
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 \(L' \in F^*\),
1.2 Covariance
Let \(F\) be a Banach space and \(E\) be a Hilbert space.
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
For \(\mu \) a measure on \(F\) with finite second moment and \(L \in F^*\), \(C_\mu (L, L) = \mathbb {V}_\mu [L]\).
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
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 \(L_y(z) = \langle y, z \rangle \).
Let \(E\) and \(F\) be two finite dimensional Hilbert spaces, \(\mu \) 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
in which \(L^\dagger : F \to E\) is the adjoint of \(L\).
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 \(u, v \in E\),
This is the covariance bilinear form \(C'_\mu (u, v)\), as a matrix.
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
in which \(L^\dagger : F \to E\) is the adjoint of \(L\).