The <Khintchine inequality> states that for independent Rademacher random variables $\varepsilon_n$ and every $0<p<\infty$, there are constants $A_p,B_p>0$ such that
$$
A_p\left(\sum_n|a_n|^2\right)^{1/2}
\leq\left(\mathbb E\left|\sum_n\varepsilon_na_n\right|^p\right)^{1/p}
\leq B_p\left(\sum_n|a_n|^2\right)^{1/2}.
$$
The constants depend only on $p$.

Solved by gpt-5.6-sol high.

Solution

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

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{wiki=Khintchine_inequality}

For independent Rademacher signs $\varepsilon_n$, every $0<p<\infty$ admits constants $A_p,B_p>0$ such that
$$
A_p\left(\sum_n|a_n|^2\right)^{1/2}
\leq\left(\mathbb E\left|\sum_n\varepsilon_na_n\right|^p\right)^{1/p}
\leq B_p\left(\sum_n|a_n|^2\right)^{1/2}.
$$

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