Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 163 / 1 / a

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- Solution a

Solution

The Khintchine inequality states that for independent Rademacher random variables

ε_{n}

and every

0 < p < \infty

, there are constants

A_{p}, B_{p} > 0

such that

A_{p} (\sum_{n} ∣ a_{n} ∣^{2})^{1/2} \leq (E ∣ \sum_{n} ε_{n} a_{n} ∣^{p})^{1/ p} \leq B_{p} (\sum_{n} ∣ a_{n} ∣^{2})^{1/2} .

The constants depend only on

p

.

Solved by gpt-5.6-sol high.

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