from independence to iid

Given $n$ independent random variables, a natural question is “how far” these are from iid random variables. In 1998, Montgomery-Smith and Pruss gave the following general result, which allows the concentration of independent observations to be upper bounded by iid observations, with only a constant factor loss.

The result is as follows: There exists a constant $c>0$ such that if $X_1,\dots ,X_n$ are independent random variables in some Banach space, then for all $\lambda >0$,

$$\mathbb{P}(\Vert S_n\Vert \ge \lambda )\le c\mathbb{P}\left(\Vert \sum _i{Y}_i\Vert \ge \lambda /c\right).$$

where $S_n={\sum }_i{X}_i$ and $Y_1,\dots ,Y_n$ are iid such that

$$\mathbb{P}(Y_i\in A)=\frac{1}{n}\sum _i\mathbb{P}(X_i\in A),$$

for all Borel sets $A$. Therefore, if one is only worried about the asymptotics of concentration, independence is equivalent to iid.