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

Table of contents
- Solution b

Solution

Independence makes cumulant-generating functions additive:

ψ_{Z} (λ) = \sum_{i = 1}^{n} ψ_{X_{i}} (λ) .

ν = \sum_{i = 1}^{n} ν_{i}, c = max_{1 \leq i \leq n} c_{i} .

For

0 < λ < 1/ c

, every denominator is positive and

1 - c_{i} λ \geq 1 - c λ

, so

ψ_{Z} (λ) \leq \sum_{i} \frac{ν _{i} λ ^{2}}{2 ( 1 - c _{i} λ )} \leq \frac{ν λ ^{2}}{2 ( 1 - c λ )} .

Also

E Z = 0

, and hence

Z \in Γ_{+} (ν, c)

.

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