Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 4 / b

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- Solution b

Solution

R_{X} (μ) = sup_{∥ v ∥_{2} \leq 1} ∣ MOM_{k} (X v) - μ^{T} v ∣.

For every unit-ball vector

v

, the triangle inequality gives

∣ v^{T} (μ - μ_{0}) ∣ \leq ∣ MOM_{k} (X v) - μ^{T} v ∣ + ∣ MOM_{k} (X v) - μ_{0}^{T} v ∣.

Taking the supremum and using optimality,

R_{X} (μ) \leq R_{X} (μ_{0})

, yields

∥ μ - μ_{0} ∥_{2} \leq R_{X} (μ) + R_{X} (μ_{0}) \leq 2 R_{X} (μ_{0}) .

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