Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 2 / b

Table of contents
- Solution b

Solution

Since

E f (X) = 0

and the moment-generating function is finite near zero,

lo g F (u) = O (u^{2})

. Consequently

2^{m} lo g F (λ / 2^{m}) = O (2^{- m}) \to 0,

which proves

F (λ / 2^{m})^{2^{m}} \to 1

.

For

λ_{*} = C_{P} (X)^{- 1/2}

, part (a) and the elementary bound

- lo g (1 - x) \leq 4 x /3

for

0 \leq x \leq 1/4

give

lo g F (λ_{*}) \leq \sum_{k \geq 0} 2^{k} [- lo g (1 - 4^{- (k + 1)})] \leq \frac{2}{3} < lo g 3.

Thus

F (λ_{*}) \leq 3

.

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