This is part of Exercise 7.1.9 from Vershynin's book "High Dimensional Probability":
Suppose $X_1(t),\dots,X_N(t)$ are $N$ independent, mean zero random processes indexed by points $t\in T$. Let $\varepsilon_1,\dots,\varepsilon_N$ be independent symmetric Bernoulli random variables (Rademacher variables). Prove that $$\frac{1}{2}\mathbb{E}\sup_{t\in T}\sum_{i=1}^N \varepsilon_i X_i(t) \leq \mathbb{E}\sup_{t\in T}\sum_{i=1}^N X_i(t)$$
By some standard symmetrization arguments (introducing ghost processes $X_i'(t)$ and subtracting their expectation in the LHS, I arrived at $$\mathbb{E}\sup_{t\in T}\sum_{i=1}^N \varepsilon_i X_i(t) \leq \mathbb{E}_{X,X'}\left[\sup_{t\in T} \sum_{i=1}^N X_i(t)-X_i'(t)\right]$$
Without any other assumption (or the presence of an absolute value) I don't think we can conclude that the LHS of the above inequality is bounded by $2\mathbb{E}\sup_{t\in T}\sum_{i=1}^N X_i(t)$. Is there any other argument that circumvents this obstacle or is it possible that this inequality does not hold without further assumptions or absolute values?
