On the quasi-arithmetic Gauss-type iteration

For a sequence of continuous, monotone functions $f_1,\dots,f_n \colon I \to $\mathbb{R}$$($I$isaninterval)wedefinethemapping$M \colon I^n \to I^n$asaCartesianproductofquasi-arithmeticmeansgeneratedby$f_j$-s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of $I^n$. We will prove that whenever all $f_j$-s are ${\mathcal{C}}^2$ with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit $\frac{\text{Var}{M}^{k+1}(v)}{(\text{Var}{M}^k(v){)}^2}$ will be calculated in a nondegenerated case.