Does Hoeffding's inequality hold for uncorrelated random variables?

Question

I know that Hoeffding's inequality holds for sums of independent random variables. However, we also know that being uncorrelated does not necessarily imply independence. But I wish to understand whether Hoeffding's inequality is valid for sums of bounded uncorrelated random variables? Or is independence a necessity?

Answer 1

If you think about the random variables $E_n = \exp(2 \pi i n X)$, where $X$ is chosen uniformly over $[0,1]$, then they are uncorrelated. But for almost any function $f:[0,1] \to \mathbb C$, the random variable $f(X)$ can be decomposed into a linear combination of the $E_n$ via the Fourier Series: $$ f(X) = \sum_{n=-\infty}^\infty c_n E_n ,$$ $$ E_n = \tfrac1{2\pi} \mathbb E(f(X) E_{-n}) .$$ Thus anything like Hoeffding's inequality, which predicts a sub-Gaussian behavior, is impossible.