A = (a nk) be a regular summability matrix. In the present paper we deal with subspaces of the space of A−statistically convergent sequences obtained by the rate at which the A−statistical limit tends to zero. We prove that a sequence is the A−strongly convergent if and only if it is the A−statistically convergent and the A−uniformly integrable with the rate of o (a n) where a = (a n) is a positive nonincreasing sequence. We also make a link between the A−strong convergence and the A−distributional convergence with the rate of o (a n). Finally, as an application we present an approximation theorem of Korovkin type.