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

Table of contents
- Solution e

Solution

Since

C_{X} ϕ_{k} = λ_{k} ϕ_{k}

and the eigenfunctions can be chosen orthonormally, independence of the sample gives

Cov (⟨ μ, ϕ_{k} ⟩, ⟨ μ, ϕ_{k^{'}} ⟩) = \frac{1}{n} ⟨ C_{X} ϕ_{k}, ϕ_{k^{'}} ⟩ = \frac{λ _{k}}{n} 1_{{k = k^{'}}} .

This remains valid for repeated eigenvalues after choosing an orthonormal eigenbasis within each eigenspace.

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