  We present a framework to calculate large deviations for nonlinear functions
of independent random variables supported on compact sets in Banach spaces, by
extending the result in Chatterjee and Dembo [6]. Previous research on
nonlinear large deviations has only focused on random variables supported on
$\{-1,+1\}^{n}$, a small subset of random objects people usually study, thus it
is of natural interest and need to research the corresponding theory for random
variables with general distributions. Since our results put fewer constraints
on the random variables, it has considerable flexibility in application. To
show this, we provide examples with continuous and high dimensional random
variables. Our framework could also be used to verify the mathematical rigor of
the mean field approximation method; to demonstrate, we verify the mean field
approximation for a class of spin vector models.
