A standard argument in empirical process theory leads to the following inequality: let $Z_1, \dots, Z_n$ be i.i.d random variables and let $g$ be a convex function. Then it holds that

$$ \mathbb{E}\left[g\left( \sum_{i=1}^n Z_i - \mathbb{E}[Z_i] \right)\right] \leq 2 \mathbb{E} \left[ g\left( \sum_{i=1}^n \varepsilon_i Z_i \right) \right] $$

where $\varepsilon_1, \dots \varepsilon_n$ are i.i.d Rademacher variables.

Question: is there a version of this argument when $Z_i$ are identically distributed but not necessarily independent?


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