  In various disordered systems or non-equilibrium dynamical models, the large
deviations of some observables have been found to display different scalings
for rare values bigger or smaller than the typical value. In the present paper,
we revisit the simpler observables based on independent random variables,
namely the empirical maximum, the empirical average, the empirical non-integer
moments or other additive empirical observables, in order to describe the cases
where asymmetric large deviations already occur. The unifying starting point to
analyze the large deviations of these various empirical observables is given by
the Sanov theorem for the large deviations of the empirical histogram : the
rate function corresponds to the relative entropy with respect to the true
probability distribution and it can be optimized in the presence of the
appropriate constraints. Finally, the physical meaning of large deviations rate
functions is discussed from the renormalization perspective.
