Limit distributions for non differentiable maps.

The invariance principle states that the empirical distribution
function (e.d.f.) when properly normalized and centered around the
distribution function (d.f.),  converges weakly to some universal
limit process. The result is applicable to i.i.d. data as well
stationary data (weakly or strongly dependent).

In many estimation problems the estimand can be seen as a function of
the d.f.. Plugging in the e.d.f. gives an estimator of the
estimand. Assuming smoothness for the function, such as functional
differentiability, implies that the resulting estimator converges
weakly. This is the topic for so called regular functional estimation,
in which case (for i.i.d. data) typically the limit process is
Gaussian and the rate of convergence is the usual parametric
square-root of n.

In contrast, when the estimand is the density function, the map is not
smooth. Assuming some form of regularity on the estimand, such as
monotonicity, still implies limit distribution results, however with
slower rates and sometimes limit vectors that are not Gaussian; this
is non regular estimation.

We present a general approach to non regular estimation, demanding
continuity, invariance and locality of the map defining the estimand
and demanding weak convergence of a smooth version of the rescaled
empirical process. This provides a general framework for previous
results on limit distributions  for the isotonic regression map (by
Anevski and Hössjer) and for the Hardy-Littlewood-Polya map (by
Anevski and Fougeres), and could be seen as an analogue to the
functional differentiability assumption in regular functional
estimation.