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Bernstein moment condition Created 2026-09-28 Updated 2026-11-05

A centered random variable

X

with

E [X^{2}] = ν

satisfies the Bernstein moment condition with parameter

b

when

∣ E [X^{k}] ∣ \leq \frac{k !}{2} ν b^{k - 2}

for every integer

k \geq 2

. Summing its moment-generating series shows that it is sub-exponential with parameters

(2 ν, 2 b)

.

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 208 / 1 / d / Solution Created 2026-09-28 Updated 2026-10-03

Because

E X = 0

, the moment-generating function series and the Bernstein moment condition imply, for

∣ λ ∣ b < 1

,

E e^{λ X} = 1 + k = 2 \sum \infty \frac{λ ^{k} E [ X ^{k} ]}{k !} \leq 1 + \frac{ν λ ^{2}}{2} k = 2 \sum \infty (∣ λ ∣ b)^{k - 2} = 1 + \frac{ν λ ^{2}}{2 ( 1 - ∣ λ ∣ b )} \leq exp (\frac{ν λ ^{2}}{2 ( 1 - ∣ λ ∣ b )}) .

For

∣ λ ∣ < 1/ (2 b)

, the denominator satisfies

1 - ∣ λ ∣ b > 1/2

, so

E e^{λ X} \leq e^{ν λ^{2}} = e^{λ^{2} (2 ν) /2} .

Thus $X$ is sub-exponential with parameters $(2 ν, 2 b)$ .

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