Before revision:  Our key result is that under weak assumptions the estimator of the functional concentrates in a L/ root(n) neighborhood of the target with deviations controlled by the Gaussian law , provided L/root(n) 0; L is the operator norm  of the functional , measuring the degree of its non-regularity, with L diverging for local functionals (or under weak identification of the global functionals) .
After revision:  Our key result is that under weak assumptions the estimator of the functional concentrates in a L/ root(n) neighborhood of the target with deviations controlled by the Gaussian law , provided L/root(n) . We require that the operator norm of the functional , measuring the degree of its non-regularity, with L diverging for local functionals (or under weak identification of the global functionals) .