Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 1 / c

Table of contents
- Solution c

Solution

The Chernoff bound gives, for

0 < λ < 1/ c

,

P (X \geq t) \leq exp {- λ t + \frac{ν λ ^{2}}{2 ( 1 - c λ )}} .

u = c t / ν

, elementary differentiation shows that the exponent is minimized at

λ_{*} = \frac{1}{c} (1 - \frac{1}{1 + 2 u}) .

Substitution gives

P (X \geq t) \leq exp {- \frac{ν}{c ^{2}} h (u)}, h (u) = 1 + u - 1 + 2 u .

Thus

h

is the Legendre transform generated by the sub-Gamma cumulant bound after its natural rescaling.

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