Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 1 / a

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- Solution a

Solution

Write

Z = X - μ

, so the covariance operator is

C_{X} h = E [⟨ Z, h ⟩ Z] .

For

g, h \in L^{2} [0, 1]

,

⟨ C_{X} h, g ⟩ = E [⟨ Z, h ⟩ ⟨ Z, g ⟩] = ⟨ h, C_{X} g ⟩,

so

C_{X}

is self-adjoint. Moreover,

⟨ C_{X} h, h ⟩ = E [⟨ Z, h ⟩^{2}] \geq 0,

so it is a positive operator.

Let

(e_{j})

be any orthonormal basis. Tonelli theorem and Parseval identity give

tr C_{X} = \sum_{j} ⟨ C_{X} e_{j}, e_{j} ⟩ = E \sum_{j} ∣ ⟨ Z, e_{j} ⟩ ∣^{2} = E ∥ Z ∥^{2} < \infty.

A positive operator with finite trace is a trace-class operator, completing the proof.

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