Iterations of mean-type mappings and their convergence

We define so-called residual means, which have a Taylor expansion of the form M(x)= x +12 ξM( x) Var(x)+o(\|x- x\|^α) for some α>2 and a single-variable function ξM (x~stands for the arithmetic mean of the vector x), and show that all symmetric means which are three times continuously differentiable are residual. We also calculate the value of residuum for quasideviation means and a few subclasses of this family. Later, we apply it to establish the limit of the sequence (Var\ Mⁿ⁺¹(x)(Var\ Mⁿ(x))²)ₙ₌₁^∞, where M Iᵖ→ Iᵖ is a mean-type mapping consisting of p-variable residual means on an interval I, and x ∈ Iᵖ is a nonconstant vector.