Starting with a data set $X_{0}$, compute its arithmetic, geometric and harmonic means, $A(X_{0}), G(X_{0})$ and $H(X_{0})$ respectively. Let $X_{1} = \{A(X_{0}),G(X_{0}),H(X_{0})\}$, and compute $A(X_{1}), G(X_{1})$ and $H(X_{1})$. Extending this, we recursively define $X_{n} = \{A(X_{n-1}),G(X_{n-1}),H(X_{n-1})\}$. Is there anything interesting about the behavior of the $\{X_{n}\}$ with regards to convergence, does a closed form representation exist, and how does it depend on the initial data set $X_{0}$?
Any insight would be appreciated. Thanks!