The main result of this article is a characterization of the permutations θ: N → N that map a set with zero asymptotic density into a set with zero asymptotic density; a permutation has this property if and only if the lower asymptotic density of Cp tends to 1 as p → ∞ where p is an arbitrary natural number and Cp = {l : θ-1(l)≤ lp}. We then show that a permutation has this property if and only if it maps statistically convergent sequences into statistically convergent sequences.

The main result of this note is that if I is an ideal generated by a regular double summability matrix summability method T that is the product of two nonnegative regular matrix methods for single sequences, then I-statistical convergence and convergence in I-density are equivalent. In particular, the method T generates a density µ_T with the additive property (AP) and hence, the additive property for null sets (APO). The densities used to generate statistical convergence, lacunary statistical convergence, and general de la Vallée-Poussin statistical convergence are generated by these types of double summability methods. If a matrix T generates a density with the additive property then T-statistical convergence, convergence in T-density and strong T-summabilty are equivalent for bounded sequences. An example is given to show that not every regular double summability matrix generates a density with additve property for null sets.