Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 2 / b

Table of contents
- Solution b

Solution

The bounded differences property with constants

c_{i}

∣ f (x) - f (x^{'}) ∣ \leq c_{i}

whenever

x

and

x^{'}

differ only in coordinate

i

. Conditional on

X_{- i}

, the range of

Z

is therefore at most

c_{i}

. The range bound on variance gives

Var (Z ∣ X_{- i}) \leq \frac{c _{i}^{2}}{4} .

Substitution into the Efron–Stein inequality yields

Var (Z) \leq \frac{1}{4} \sum_{i = 1}^{n} c_{i}^{2} .

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