Given 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 such that if are independent random variables in some Banach space, then for all ,

where and are iid such that

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

Maximal inequalities

There’s a similar result but for maximal inequalities. If are arbitrary random variables and are independent with for all , then

This is fairly straightforward to prove with a union bound and some elementary inequalities on .